Tag: ml

755 topic(s)

Multimodal Secure AlignmentMultimodal secure alignment is the problem of making a model's safety behavior consistent across text, images, audio, and mixed-modal inputs. It matters because a model can reconstruct harmful intent across modalities or through images that evade text-only filters, so defenses must align the fused system rather than just one input channel.
Layer Dropping and Progressive Pruning (TrimLLM)Layer dropping and progressive pruning reduce inference cost by cutting transformer depth rather than shrinking every matrix. TrimLLM does this progressively for domain-specialized LLMs, exploiting the empirical fact that not all layers are equally important in a target domain and aiming to retain in-domain accuracy while reducing latency.
Test-Time Compute ScalingTest-time compute scaling improves a model by spending extra computation at inference time, for example through search, verification, reranking, or adaptive refinement, instead of only scaling pretraining. It is most useful on prompts where the base model already has some chance of success, because additional compute can then amplify that success more efficiently than a much larger one-shot model.
GPT-3 & Few-Shot In-Context LearningGPT-3 showed that a 175B-parameter autoregressive Transformer can perform many tasks from natural-language instructions and a few demonstrations in the prompt, without gradient updates or task-specific fine-tuning. That result made in-context learning a central paradigm and showed that scale alone could unlock strong few-shot behavior.
The Bitter LessonThe Bitter Lesson is Sutton's argument that, over the long run, general methods that scale with compute and data outperform systems built around hand-crafted domain knowledge. It is a historical pattern claim, not a theorem, and its force comes from repeated examples in search, game playing, vision, and language.
Constitutional Classifiers++Constitutional Classifiers++ is a production-oriented jailbreak defense that uses context-aware classifiers and a cascade of cheap and expensive checks to block harmful exchanges efficiently. The system is designed to keep refusal rates and serving cost low while still catching universal jailbreaks that earlier, response-only filters missed.
Continuous Thought Machines (CTM)Continuous Thought Machines are models that make neural timing and synchronization part of the representation, instead of treating layers as purely instantaneous mappings. They use neuron-level temporal processing and support adaptive compute, so the same model can stop early on easy inputs or continue reasoning on harder ones.
Mechanistic OOCR Steering VectorsMechanistic OOCR steering vectors are a proposed explanation for some out-of-context reasoning results: fine-tuning can act like adding an approximately constant steering direction to the residual stream, rather than learning a deeply conditional new algorithm. That helps explain why a tuned behavior can generalize far beyond the fine-tuning data and why injecting or subtracting the vector can often reproduce or remove it.
Critical Representation Fine-Tuning (CRFT)Critical Representation Fine-Tuning (CRFT) is a PEFT method that improves reasoning by editing a small set of causally important hidden states instead of updating model weights broadly. It identifies critical representations through information-flow analysis and learns low-rank interventions on those states while keeping the base model frozen.
Chain-of-Thought MonitorabilityChain-of-thought monitorability is the safety claim that when a model needs explicit reasoning to complete a task, its written chain of thought can be monitored for harmful intent or deception. The key property is monitorability rather than perfect faithfulness: hiding the reasoning tends to become harder when the reasoning itself is load-bearing for success.
ZeRO (Zero Redundancy Optimizer)ZeRO (Zero Redundancy Optimizer) partitions optimizer states, gradients, and eventually parameters across data-parallel workers so each GPU no longer stores a full copy of the training state. This cuts memory dramatically and makes very large-model training feasible without requiring full model-parallel architectures.
T5 (Text-to-Text Transfer Transformer)T5 is an encoder-decoder Transformer that casts every NLP task as text-to-text generation, so translation, question answering, classification, and even some regression tasks share the same model and loss. Its span-corruption pretraining on C4 made it a landmark demonstration of unified transfer learning.
GPT-2 & Zero-Shot Task TransferGPT-2 showed that a large decoder-only language model can perform many tasks in the zero-shot setting by continuing a task-formatted prompt rather than being fine-tuned. The key result was that scale and diverse web text made translation, summarization, and question answering look like ordinary next-token prediction.
GPT-1 (Generative Pre-Training)GPT-1 established the pretrain-then-fine-tune recipe for Transformers: first train a decoder on unlabeled text with a language-model objective, then adapt it to downstream tasks with minimal task-specific layers. This showed that generic generative pretraining could beat many bespoke NLP architectures on downstream benchmarks.
ELMo (Embeddings from Language Models)ELMo produces contextualized word embeddings by taking a learned task-specific combination of hidden states from a pretrained bidirectional LSTM language model. Unlike static embeddings such as word2vec or GloVe, it gives the same word different vectors in different sentence contexts.
Sparsely-Gated Mixture of Experts (MoE)A sparsely-gated Mixture of Experts (MoE) layer routes each token to only a small subset of expert networks, so model capacity can grow much faster than compute per token. Its central challenge is routing and load balancing: without auxiliary losses, a few experts tend to monopolize traffic.
Key-Value Memory NetworksKey-Value Memory Networks store each memory slot as a key for retrieval and a separate value for the returned content. This decouples matching from payload and is a direct conceptual precursor to modern query-key-value attention.
Luong Attention (Global and Local)Luong attention is a sequence-to-sequence attention mechanism that scores decoder states against encoder states using multiplicative forms such as dot or bilinear attention. It distinguishes global attention over all source positions from local attention over a predicted window, helping make neural machine translation more scalable.
GloVe Word EmbeddingsGloVe learns word embeddings by fitting vector dot products to the log of global word-word co-occurrence counts. Because it is trained on ratios of co-occurrence statistics, linear relations such as king minus man plus woman approximately equals queen often emerge in the embedding space.
Neural Turing Machine (NTM)A Neural Turing Machine augments a neural controller with a differentiable external memory that it can read from and write to using soft attention over memory locations. It was an early attempt to learn algorithm-like behavior such as copying and sorting while remaining trainable end to end.
Xavier/Glorot InitializationXavier or Glorot initialization chooses weight variance from fan-in and fan-out so activations and gradients stay roughly stable across deep layers. It is well suited to symmetric activations such as tanh, while ReLU networks usually prefer He initialization.
ImageNet DatasetImageNet is a large, hierarchically labeled image dataset whose 1000-class ILSVRC benchmark became the defining testbed for modern computer vision. AlexNet's 2012 win on ImageNet triggered the deep learning shift by showing that GPU-trained CNNs could dramatically beat hand-engineered pipelines.
Neural Probabilistic Language ModelThe Neural Probabilistic Language Model replaced count-based n-grams with learned word embeddings and a neural network that predicts the next word from a continuous representation of context. Its core contribution was showing that distributed representations let language models generalize to unseen but similar word sequences.
Weight TyingWeight tying uses the same matrix for token embeddings and the output softmax projection, typically by setting the output weights to the transpose of the input embedding table. This cuts parameters and often improves language modeling by forcing input and output token representations to share geometry.
Gradient Checkpointing (Activation Recomputation)Gradient checkpointing saves memory by storing only selected activations during the forward pass and recomputing the missing ones during backpropagation. The trade-off is extra compute for lower peak memory, which is why it is widely used to train large Transformers that would otherwise not fit in GPU memory.
PagedAttentionPagedAttention stores the KV cache in fixed-size non-contiguous blocks, like virtual-memory pages, instead of requiring one contiguous allocation per sequence. This largely removes fragmentation, enables prompt-prefix sharing, and is a key reason vLLM can serve many more concurrent requests.
Speculative DecodingSpeculative decoding speeds up autoregressive generation by letting a small draft model propose several tokens and then having the large target model verify them in parallel. With the rejection-sampling correction from the original algorithm, the output distribution remains exactly the same as sampling from the target model alone.
KL-Divergence Penalty in RLHFThe KL-divergence penalty in RLHF keeps the learned policy close to a reference model while it maximizes reward, usually by subtracting a term proportional to the KL divergence from the objective. This stabilizes training and reduces reward hacking by discouraging the policy from drifting too far from fluent supervised behavior.
Proximal Policy Optimization (PPO)Proximal Policy Optimization is a policy-gradient algorithm that improves a policy while clipping how far action probabilities can move from the previous policy in one update. In RLHF it is usually paired with a KL penalty so the model gains reward without drifting too far from a reference model.
Next-Token Prediction Objective (Causal Language Modeling)Next-token prediction trains a causal language model to assign high probability to each token given all previous tokens. Maximizing this likelihood over large text corpora teaches the model syntax, facts, and reusable patterns that later support prompting and generation.
AdamW OptimizerAdamW is Adam with decoupled weight decay: parameter shrinkage is applied directly to the weights instead of being mixed into the adaptive gradient update. This preserves the intended regularization effect and is why AdamW became the default optimizer for many Transformer models.
SwiGLU Activation FunctionSwiGLU is a gated feed-forward activation that multiplies one linear projection by a Swish-activated gate from another projection. It usually performs better than standard ReLU-style MLP blocks at similar scale, which is why many modern LLMs use it in their feed-forward layers.
Pre-Norm vs. Post-Norm ArchitecturePre-Norm vs. Post-Norm is the choice of whether layer normalization is applied before or after each residual sublayer in a Transformer block. Pre-Norm usually trains deeper stacks more stably by preserving gradient flow through the residual path, while Post-Norm was the original design and can be less stable at scale.
Byte Pair Encoding (BPE)Byte Pair Encoding is a subword tokenization method that repeatedly merges the most frequent adjacent symbols in a corpus. It builds a vocabulary between characters and whole words, which handles rare words better than word-level tokenization while keeping sequence lengths manageable.
Softmax TemperatureSoftmax temperature rescales logits before softmax to control randomness in the output distribution. Lower temperature makes probabilities sharper and decoding more deterministic, while higher temperature flattens the distribution and increases diversity.
Key-Value (KV) CachingKey-value caching stores the attention keys and values from earlier tokens during autoregressive decoding so they do not need to be recomputed at every step. It speeds up generation dramatically, but the cache grows with sequence length and turns inference into a memory-management problem.
Rotary Positional Embedding (RoPE)Rotary Positional Embedding encodes position by rotating query and key vectors with token-index-dependent angles before attention is computed. Because the resulting dot products depend on relative offsets, RoPE gives Transformers a simple and widely used way to represent order.
Sinusoidal Positional EncodingSinusoidal positional encoding adds fixed sine and cosine patterns of different frequencies to token embeddings so the model can infer token order. The encoding is deterministic and smooth across positions, which let the original Transformer represent position without learning a separate table.
Causal (Masked) Self-AttentionCausal masked self-attention is self-attention with a mask that prevents each position from attending to future tokens. Applying the mask before softmax enforces autoregressive order, so the model can predict the next token without seeing the answer in advance.
Message Passing in Graph Neural Networks (GNNs)Message passing in graph neural networks updates each node by aggregating transformed information from its neighbors and combining it with the node's current representation. After K rounds, a node's state depends on its K-hop neighborhood, which is why message passing is the core operation of most spatial GNNs.
Triplet Margin LossTriplet margin loss trains an embedding space so an anchor is closer to a positive example than to a negative example by at least a fixed margin. It is a standard metric-learning objective because it directly enforces relative similarity rather than predicting class labels.
Lagrange MultipliersLagrange multipliers solve constrained optimization problems by introducing auxiliary variables that encode the constraints inside a single objective. At a constrained optimum, the gradient of the objective lies in the span of the constraint gradients, which is why the method is central to duality and SVM derivations.
Ordinary Least Squares (OLS) Closed-Form SolutionThe OLS closed-form solution is the exact least-squares answer computed directly from the design matrix rather than by iterative optimization. In the full-rank case it solves the normal equations, and geometrically it projects the target vector onto the column space of the features.
PAC Learning (Probably Approximately Correct)PAC learning formalizes what it means for a hypothesis class to be learnable: with enough samples, an algorithm should return a hypothesis whose error is small with high probability. It is foundational because sample complexity and model capacity can then be expressed as rigorous guarantees instead of heuristics.
Receiver Operating Characteristic (ROC) & AUCThe ROC curve plots true positive rate against false positive rate as a binary classifier's threshold changes, and AUC summarizes that curve into a single number. AUC also has a ranking interpretation: it is the probability that a random positive example scores above a random negative one.
Hidden Markov Model (HMM)A Hidden Markov Model is a sequence model with an unobserved Markov chain of states and an observed emission distribution from each state. It became a standard model for speech, tagging, and other structured sequence tasks because dynamic programming can efficiently infer likely states and sequence probabilities.
Convex FunctionA convex function is one whose value on any line segment lies below the chord connecting that segment's endpoints. This matters in optimization because convex problems have no spurious local minima: every local minimum is global.
Shannon EntropyShannon entropy measures the expected surprisal of a random variable and quantifies how uncertain its outcomes are. It is the basic information-theoretic quantity from which cross-entropy, KL divergence, mutual information, and many ML loss functions are built.
L1 vs. L2 NormsThe L1 norm sums absolute values and tends to promote sparsity when used as a penalty, while the L2 norm measures Euclidean length and tends to shrink weights smoothly without zeroing many of them out. That difference is why L1 is associated with feature selection and L2 with stable shrinkage.
K-Means Objective Function (Inertia)The K-means objective, also called inertia, is the sum of squared distances from each point to its assigned cluster centroid. K-means greedily lowers that objective by alternating between reassigning points and recomputing centroids, though the result still depends on initialization because the problem is nonconvex.
The Curse of DimensionalityThe curse of dimensionality is the collection of high-dimensional effects that make data sparse, neighborhoods less informative, and sample requirements explode as dimension grows. It helps explain why distance-based methods, density estimation, and exhaustive search often break down in large feature spaces.
Markov Chain Monte Carlo (MCMC)Markov Chain Monte Carlo samples from a difficult target distribution by constructing a Markov chain whose stationary distribution matches that target. It is essential in Bayesian inference because it replaces intractable posterior integrals with averages over samples, provided the chain mixes well enough.
Reparameterization Trick (VAE)The reparameterization trick writes a stochastic latent sample as a differentiable transformation of parameters and noise, typically z equals mu plus sigma times epsilon. This lets gradients flow through sampling and makes variational autoencoder training practical with backpropagation.
AutoencodersAutoencoders are neural networks trained to reconstruct their inputs after passing them through a compressed or otherwise constrained latent representation. They are useful because the bottleneck forces the model to learn structure in the data rather than just memorize an identity map.
GAN Minimax ObjectiveThe GAN minimax objective sets up a two-player game in which a generator tries to produce samples that fool a discriminator, while the discriminator tries to distinguish real from generated data. At equilibrium the generator matches the data distribution, though the training game is often unstable in practice.
Q-LearningQ-learning is an off-policy reinforcement learning algorithm that learns the optimal action-value function by bootstrapping from a Bellman target over the best next action. Because its update does not require following the current policy, it became a foundational method in both tabular RL and DQN-style deep RL.
The Bellman EquationThe Bellman equation recursively expresses the value of a state or state-action pair as immediate reward plus discounted expected future value. It is the backbone of dynamic programming and reinforcement learning because it turns long-horizon return into a local consistency condition.
The Markov PropertyThe Markov property says that the conditional distribution of the future depends only on the present state, not on the full past history, once the state is known. It is the defining assumption behind Markov chains and MDPs and tells you when a state representation is sufficient for planning.
Bootstrap Aggregating (Bagging)Bootstrap Aggregating trains multiple models on bootstrap-resampled versions of the training set and averages their predictions to reduce variance. It helps most with unstable base learners such as decision trees, which is why it underlies random forests.
Gradient Boosting Machines (GBM)Gradient Boosting Machines build an additive model by fitting each new weak learner to the negative gradient of the current loss. In practice each stage focuses on correcting the remaining errors of the ensemble, which makes boosting powerful but sensitive to overfitting if trees and learning rates are not controlled.
The Fisher Information MatrixThe Fisher Information Matrix measures how sensitive a model's log-likelihood is to changes in its parameters and therefore captures local statistical curvature. It underlies asymptotic variance bounds and natural gradient methods because it defines a geometry tied to the model's predictive distribution.
AdaBoost (Adaptive Boosting)AdaBoost builds an ensemble by repeatedly fitting weak learners to reweighted data so that previously misclassified examples receive more attention. Its final predictor is a weighted vote of the learners, and its power comes from turning many slightly better-than-random classifiers into a strong one.
Information Gain (Decision Trees)Information gain is the reduction in entropy achieved by splitting a dataset on a candidate feature. Decision-tree algorithms use it to choose splits that most reduce label uncertainty, though raw information gain can be biased toward features with many distinct values.
DropoutDropout regularizes a neural network by randomly zeroing activations during training, which prevents units from co-adapting too strongly. At test time the full network is used with rescaled activations, making dropout behave like an inexpensive ensemble-style regularizer.
Singular Value Decomposition (SVD)Singular Value Decomposition factors any matrix into orthogonal directions and nonnegative singular values. It is fundamental because low-rank approximation, PCA, pseudoinverses, compression, and many denoising methods all follow from that decomposition.
Maximum A Posteriori (MAP) EstimationMaximum a posteriori estimation chooses the parameter value that maximizes posterior probability given the data. It is equivalent to maximum likelihood plus a log-prior regularizer, which is why MAP connects Bayesian estimation to familiar penalized optimization objectives.
Supervised learningSupervised learning trains a model on labeled input-output pairs so it can predict the correct target on new examples from the same distribution. Classification and regression are its two main forms, depending on whether the target is discrete or continuous.
Unsupervised learningUnsupervised learning tries to discover structure in data without labeled targets, such as clusters, latent factors, or a density model. It is used for representation learning, dimensionality reduction, clustering, and generative modeling when explicit supervision is unavailable.
Reinforcement learning (RL)Reinforcement learning studies how an agent should act through trial and error to maximize cumulative reward in an environment. Unlike supervised learning, feedback is delayed and depends on the agent's own actions, so the problem is about sequential decision-making as much as prediction.
Linear functionA linear function satisfies additivity and homogeneity, so it can be written as a matrix map with no bias term. In machine learning people often use 'linear' loosely for affine maps, but mathematically the distinction matters because adding a bias breaks true linearity.
Affine transformationAn affine transformation is a linear map followed by a translation, so it has weights and a bias. Dense neural network layers are affine rather than strictly linear, because the bias lets the model shift activations and decision boundaries.
Loss functionA loss function maps a model's prediction and the true target to a scalar error signal that training aims to minimize. It defines what the model is optimized for, so changing the loss changes which mistakes are treated as costly.
Mean squared error (MSE)Mean squared error averages the squared difference between predicted and true values, making large errors count disproportionately more than small ones. For regression it is especially important because minimizing MSE is equivalent to maximum likelihood under Gaussian noise.
Prediction errorPrediction error is the difference between a model's prediction and the true target for an example. It is the atomic quantity from which losses, residual analysis, and generalization metrics are built.
Gradient descentGradient descent minimizes a differentiable objective by repeatedly moving parameters in the direction of steepest local decrease, namely the negative gradient. Its step size is set by the learning rate, so convergence depends on both objective geometry and update scale.
Mini-batch gradient descentMini-batch gradient descent estimates the gradient on a small subset of training examples at each update instead of on the full dataset or a single example. It is the practical default in deep learning because it balances hardware efficiency with optimization noise.
Stochastic gradient descent (SGD)Stochastic gradient descent updates parameters using a gradient estimate from one example or a very small random batch, making each step noisy but cheap. That noise can slow exact convergence yet often helps large models optimize and generalize in practice.
ConvergenceIn optimization, convergence means an algorithm's iterates approach a stable solution or stationary point as updates continue. In practice people often mean that the loss or parameters stop changing much, though true convergence depends on the objective and algorithmic assumptions.
GeneralizationGeneralization is a model's ability to perform well on unseen data from the same underlying distribution as its training data. It is the real goal of learning, because low training error alone can come from memorization rather than useful structure.
RegularizationRegularization is any technique that biases learning toward simpler, more stable, or less overfit solutions. It can appear as an explicit penalty such as weight decay or as an implicit training choice such as data augmentation, dropout, or early stopping.
L1 regularization (Lasso)L1 regularization adds a penalty proportional to the sum of absolute parameter values, encouraging many coefficients to become exactly zero. That sparsity makes Lasso useful when feature selection is part of the goal, not just shrinkage.
L2 regularization (Ridge/Weight Decay)L2 regularization adds a penalty proportional to the sum of squared parameter values, shrinking weights toward zero without usually making them exactly sparse. In plain SGD it is equivalent to weight decay and is widely used because it improves stability and reduces variance.
Early stoppingEarly stopping regularizes training by halting optimization when validation performance stops improving and keeping the best checkpoint seen so far. It works because prolonged optimization can eventually fit noise or idiosyncrasies of the training set rather than signal.
HyperparameterA hyperparameter is a setting chosen outside the optimization loop, such as learning rate, model width, regularization strength, or batch size. Unlike learned parameters, hyperparameters govern how the model is trained or structured and are usually selected by validation.
Data leakageData leakage occurs when information that would not be available at prediction time leaks into training or model selection, causing overly optimistic evaluation. Common examples are fitting preprocessing on the full dataset, peeking at test labels, or using future information in time-series tasks.
BackpropagationBackpropagation computes gradients of a scalar loss with respect to all network parameters by applying the chain rule backward through the computation graph. It makes deep learning practical because it turns a complicated nested function into reusable local gradient calculations.
Forward passThe forward pass is the computation that maps input data through the model to produce activations and an output prediction. During training it also caches intermediate values needed later by the backward pass.
Backward passThe backward pass propagates gradients from the loss back through the computation graph to determine how each parameter affected the final error. It uses stored forward-pass intermediates and the chain rule to accumulate derivatives efficiently.
Chain ruleThe chain rule gives the derivative of a composition of functions by multiplying local derivatives along the computation path. It is the mathematical principle that backpropagation applies at scale throughout a neural network.
Automatic differentiationAutomatic differentiation computes exact derivatives of a program by systematically composing derivatives of its primitive operations. Unlike symbolic differentiation it does not manipulate formulas, and unlike numerical differentiation it does not rely on finite-difference approximations.
Computational graphA computational graph represents a calculation as nodes for variables or operations and edges for data dependencies. It is useful because the same graph that defines the forward computation can also be traversed backward to perform automatic differentiation.
Neural networkA neural network is a parameterized function built by composing affine transformations with nonlinear activations across layers. Its power comes from learning representations from data rather than relying on hand-crafted features for each task.
Learning rateThe learning rate is the scalar that sets how large each optimization step is when parameters are updated. If it is too high training can diverge or oscillate, and if it is too low training can become extremely slow or get stuck.
EpochAn epoch is one complete pass through the training dataset. In mini-batch training, an epoch consists of many updates, one for each batch needed to cover the data once.
OverfittingOverfitting happens when a model fits patterns specific to the training set, including noise, better than it captures the underlying data-generating structure. The usual symptom is low training error paired with substantially worse validation or test error.
UnderfittingUnderfitting happens when a model is too limited, too constrained, or too poorly trained to capture the main structure in the data. It usually shows up as high error on both training and validation data, indicating high bias rather than variance.
NeuronA neuron in a neural network computes a weighted sum of its inputs, adds a bias, and applies an activation function. Collections of neurons form layers, so a single neuron's role is simple even though many together can represent complex functions.
Activation functionAn activation function is the nonlinear mapping applied after an affine transformation in a neural network. It is what prevents a stack of layers from collapsing into one affine map, enabling deep networks to approximate complex functions.
SigmoidThe sigmoid function maps a real number to a value between zero and one, making it easy to interpret as a probability or gate. Its downside is saturation at large positive or negative inputs, which can cause vanishing gradients in deep networks.
TanhThe tanh function maps inputs to the range minus one to one and is zero-centered, which often makes optimization easier than with the sigmoid. Like the sigmoid, however, it still saturates at large magnitudes and can cause vanishing gradients.
ReLUReLU outputs the positive part of its input and zero otherwise. It became the default activation in many deep networks because it is simple, cheap, and far less prone to saturation than sigmoid or tanh, though units can still die if they stay on the zero side.
SoftmaxSoftmax turns a vector of logits into a probability distribution by exponentiating and normalizing them so the components sum to one. It is commonly used for multiclass prediction because it converts arbitrary scores into class probabilities while preserving their ranking.
Fully connected layerA fully connected layer applies an affine transformation in which every output unit depends on every input feature. It is the standard dense layer used in multilayer perceptrons and as a projection block inside many larger architectures.
Input layerThe input layer is the entry point of a network, where raw or preprocessed features are presented to the model. Unlike hidden layers, it usually performs little or no learned computation by itself and mainly defines the representation the rest of the network receives.
Hidden layerA hidden layer is any internal layer between the input and output of a network. Hidden layers transform raw inputs into increasingly useful intermediate representations that the final output layer can read out.
Output layerThe output layer is the final transformation that maps a model's last hidden representation to a prediction space such as class logits, probabilities, or regression values. Its shape and activation depend on the task being solved.
Multilayer perceptron (MLP)A multilayer perceptron is a feedforward neural network made of stacked fully connected layers and nonlinear activations. It is the canonical dense architecture for tabular function approximation and the feed-forward subnetwork inside many Transformer blocks.
Deep neural networkA deep neural network is a neural network with multiple hidden layers rather than just one or two. The extra depth lets it build hierarchical features and represent complex functions more efficiently than shallow networks in many settings.
Deep learningDeep learning is the study and practice of training multilayer neural networks on data to learn useful representations automatically. Its hallmark is end-to-end learning of features and predictors together, especially when large data and compute are available.
Composite functionA composite function applies one function to the output of another, such as f of g of x. Neural networks are composite functions at scale, which is why gradients are computed by repeatedly applying the chain rule.
Feedforward neural networkA feedforward neural network is a network whose computations move from input to output without recurrent cycles. Each layer depends only on earlier activations in the same pass, making feedforward networks the basic template for MLPs and many vision models.
Convolutional neural network (CNN)A convolutional neural network uses learned convolution filters with local receptive fields and weight sharing to process grid-like data such as images. Those inductive biases make CNNs especially effective and parameter-efficient for visual pattern recognition.
Recurrent neural network (RNN)A recurrent neural network processes sequences by maintaining a hidden state that is updated one step at a time from the current input and previous state. This gives it a notion of temporal memory, but plain RNNs are hard to train on long dependencies because gradients can vanish or explode.
Elman RNNAn Elman RNN is the classic simple recurrent network in which the next hidden state is a nonlinear function of the current input and previous hidden state. It introduced the basic hidden-state recurrence used by later gated models, but long-range memory is poor without gating.
Hidden stateThe hidden state is the internal representation a sequential model carries forward as it processes inputs over time. In an RNN or LSTM it summarizes relevant past context, and in broader neural architectures it usually means a layer's intermediate activation vector.
Backpropagation through time (BPTT)Backpropagation through time trains a recurrent network by unrolling it across sequence steps and applying backpropagation to the resulting deep computational graph. It exposes how earlier states influence later losses, but long unrolls make optimization and memory use difficult.
What is the vanishing gradient problem?The vanishing gradient problem is the tendency for gradients propagated through many layers or time steps to shrink exponentially, making early parameters learn extremely slowly. It is especially severe in deep sigmoid or tanh networks and was a main motivation for LSTMs, better initialization, and residual connections.
Long short-term memory (LSTM)Long short-term memory is a gated recurrent architecture designed to preserve information over long timescales. Its input, forget, and output gates regulate a cell state with near-linear self-connections, which helps prevent the vanishing-gradient behavior of simple RNNs.
xLSTMxLSTM is a family of modern LSTM variants that adds exponential gating and redesigned memory structures, including scalar-memory and matrix-memory forms, to make recurrent models more scalable. The goal is to keep LSTM-style recurrence while improving stability, parallelism, and long-context performance.
minLSTMminLSTM is a simplified LSTM variant designed to remove some of the sequential dependencies that make classical LSTMs expensive while keeping useful gating behavior. The result is a lighter recurrent block that can be trained more efficiently and scaled more easily.
Gated recurrent unit (GRU)A gated recurrent unit is a recurrent architecture that uses update and reset gates to control how much past information is kept and how much new input is written into the hidden state. It is simpler than an LSTM because it has no separate cell state, yet it often achieves similar sequence-modeling performance.
Embedding layerAn embedding layer maps discrete IDs such as words, subwords, or items to learned dense vectors. It is essential whenever symbolic inputs must be represented in a continuous space that gradient-based models can manipulate.
Embedding vectorAn embedding vector is the dense continuous representation assigned to a discrete token, item, or entity by an embedding table or model. Its meaning comes from geometry: similar entities tend to occupy nearby directions or neighborhoods in the learned space.
Word embeddingA word embedding is a dense vector representation of a word learned from distributional context rather than hand-coded features. Its purpose is to place semantically or syntactically related words near one another in vector space so downstream models can generalize across vocabulary items.
Word2VecWord2Vec is a family of shallow neural methods that learn word embeddings from local context, most famously via the skip-gram and CBOW objectives. Its importance is that simple predictive training on large text corpora produced useful semantic geometry, including analogy-like linear regularities.
Skip-gramSkip-gram trains a model to predict surrounding context words from a center word. It learns embeddings that are especially good for capturing rare-word semantics because each observed word directly becomes a prediction source for many context targets.
FastTextFastText extends Word2Vec by representing a word as a bag of character n-gram embeddings rather than as a single atomic vector. That lets it model morphology and produce reasonable embeddings for rare or even unseen words.
Semantic similaritySemantic similarity is the degree to which two words, sentences, or documents share meaning rather than just surface form. In machine learning it is often estimated with embeddings and cosine similarity, which turns meaning comparison into a geometric problem.
Cosine similarityCosine similarity measures the angle between two vectors: \( \cos \theta = x \cdot y / (\|x\| \|y\|) \). It ignores magnitude and compares direction, which is why it is the default similarity metric for embeddings in retrieval, clustering, and semantic search.
Bag of wordsBag of words represents a document by counts or weights of vocabulary terms while discarding word order and syntax. It is simple, sparse, and historically central to information retrieval and document classification, but it cannot distinguish sentences with the same words in different orders.
Document-term matrixA document-term matrix is a matrix whose rows are documents, columns are vocabulary terms, and entries are counts or weights such as TF-IDF. It is the core data structure behind bag-of-words retrieval, topic modeling, and many classical NLP pipelines.
TF-IDFTF-IDF weights a term by how frequent it is in a document and how rare it is across the corpus, typically \( tf(w,d) \log(N/df(w)) \). It downweights ubiquitous words and highlights terms that are especially informative for a given document.
SparsitySparsity means most entries in a vector, matrix, or parameter set are exactly zero. In ML it matters because sparse representations save memory and computation, and because sparsity-inducing penalties such as L1 can make models more interpretable.
Dense vectorA dense vector is a low- or moderate-dimensional representation in which most entries are nonzero. Dense vectors are usually learned embeddings, so they capture semantic similarity better than sparse count vectors but are harder to interpret directly.
Sparse vectorA sparse vector has very few nonzero entries relative to its dimensionality. Classical text features such as bag-of-words and TF-IDF are sparse, which makes them memory-efficient and interpretable even when the feature space is huge.
One-hot encodingOne-hot encoding represents a categorical variable as a binary vector with exactly one 1 and all other entries 0. It preserves category identity without implying any ordering, but its dimensionality grows linearly with the number of categories.
TokenizationTokenization is the process of splitting raw text into model-readable tokens such as words, subwords, bytes, or characters. It determines vocabulary size, sequence length, and how efficiently a language model handles rare words, multilingual text, and code.
TokenA token is the discrete unit a language model reads and predicts. Depending on the tokenizer, a token may be a word, subword, byte, punctuation mark, or special control symbol, and token count determines both context usage and API cost.
SubwordA subword is a token unit smaller than a full word but larger than a character, learned to balance vocabulary size against sequence length. Subword tokenization lets models handle rare and novel words by composing them from reusable pieces.
VocabularyA vocabulary is the fixed set of tokens a tokenizer can map text into and a model can natively process. Its size trades off compression against flexibility: larger vocabularies shorten sequences, while smaller ones rely more on subword or byte composition.
CorpusA corpus is a structured collection of text used to train, fine-tune, or evaluate language models. Its size, quality, domain mix, and cleaning decisions strongly shape what a model knows, how it generalizes, and which biases it inherits.
N-gramAn n-gram is a contiguous sequence of \( n \) tokens, such as a bigram for \( n=2 \) or trigram for \( n=3 \). N-grams are the basic units of classical language models and many text features because they capture short-range local context.
Count-based language modelA count-based language model estimates sequence probabilities from n-gram counts in a corpus, then uses smoothing or backoff for unseen events. It was the dominant pre-neural approach to language modeling, but it struggles with long context and data sparsity.
Language modelA language model assigns probabilities to token sequences, or equivalently predicts missing or next tokens from context. This unifies classical n-gram models, masked models like BERT, and autoregressive LLMs such as GPT under one probabilistic framework.
Autoregressive language modelAn autoregressive language model generates text left-to-right by modeling \( P(w_t \mid w_{<t}) \) for each token. Because it only conditions on past tokens, it can be used directly for open-ended generation as well as scoring sequences.
Masked language modelA masked language model is trained to recover tokens hidden within a sequence using both left and right context. This bidirectional training makes MLMs strong encoders for understanding tasks, but less natural than autoregressive models for direct generation.
Causal language modelA causal language model predicts each token using only earlier tokens, enforced by a causal attention mask. It is essentially the same modeling family as an autoregressive language model, with the word 'causal' emphasizing the masking constraint in self-attention.
Chat language modelA chat language model is a pretrained LLM further tuned to follow instructions and handle multi-turn dialogue. It is usually built by supervised fine-tuning plus preference optimization or RLHF, so it behaves more helpfully and safely than the raw base model.
PerplexityPerplexity is the exponentiated average negative log-likelihood of a test sequence, so lower perplexity means the model is less surprised by the data. It is a standard intrinsic metric for language models, though low perplexity does not guarantee downstream usefulness.
Log-likelihoodLog-likelihood is the logarithm of the probability a model assigns to the observed data under given parameter values. Taking logs turns products into sums, making estimation numerically stable and turning maximum likelihood into a tractable optimization problem.
Negative log-likelihoodNegative log-likelihood is the loss obtained by negating the log-likelihood, so maximizing probability becomes minimizing a positive objective. It is the standard training loss for probabilistic classifiers, language models, and many generative models.
Maximum likelihood estimate (MLE)The maximum likelihood estimate selects the parameter values that make the observed data most probable under the model. Many standard ML objectives, including cross-entropy for classification and next-token prediction for LLMs, are just MLE written as minimization of negative log-likelihood.
Laplace smoothingLaplace smoothing adds a small constant, often 1, to every discrete count before normalizing probabilities. It prevents zero-probability events in models such as naive Bayes and n-gram LMs, though it can over-smooth when the vocabulary is large.
Conditional probabilityConditional probability is the probability of an event after restricting attention to cases where another event is known to occur, written \( P(A \mid B) = P(A,B)/P(B) \). It is the basic object behind Bayes' rule, autoregressive models, and all context-dependent prediction.
Discrete probability distributionA discrete probability distribution assigns nonnegative probabilities to a countable set of outcomes that sum to 1. Softmax outputs in classification and next-token prediction are discrete distributions over labels or vocabulary items.
Backoff (N-gram backoff)Backoff is an n-gram smoothing strategy that uses a high-order estimate when it has enough evidence and otherwise falls back to a lower-order n-gram. It handles sparsity by preferring specific context when available without assigning zero probability to unseen sequences.
Zipf's lawZipf's law says a word's frequency is roughly inversely proportional to its rank in the frequency table. This heavy-tailed structure explains why a few tokens dominate corpora, why vocabularies keep growing with more data, and why tokenization and smoothing are central in NLP.
Cross-entropyCross-entropy measures the average coding cost of samples from a true distribution \( p \) when encoded using a model distribution \( q \). In ML it is the standard loss for classification and language modeling, and minimizing it is equivalent to maximum likelihood up to an entropy constant.
Binary cross-entropyBinary cross-entropy is the cross-entropy loss for a Bernoulli target, typically \( -[y\log \hat p + (1-y)\log(1-\hat p)] \). It is the standard loss for binary classification and for multi-label problems where each label is predicted independently.
LogitA logit is the raw score before sigmoid or softmax normalization. In binary settings, the logit is also the log-odds \( \log\frac{p}{1-p} \), which is why linear models such as logistic regression operate naturally in logit space.
ClassificationClassification is a supervised learning task in which the target is a discrete label rather than a continuous value. The model learns decision boundaries that separate classes, often outputting calibrated class probabilities as well as the predicted label.
Binary classificationBinary classification is classification with exactly two classes, usually framed as predicting the probability of a positive class. It is commonly trained with logistic regression or a sigmoid output and binary cross-entropy loss.
Multiclass classificationMulticlass classification assigns each input to exactly one of \( K>2 \) mutually exclusive classes. Models usually produce a softmax distribution over classes and train with cross-entropy against a one-hot or label-smoothed target.
RegressionRegression is a supervised learning task where the target is continuous rather than categorical. The model predicts a numeric value, and common losses such as mean squared error correspond to assumptions about the noise model, especially Gaussian noise.
Linear RegressionLinear regression models a target as an affine function of the inputs, typically \( y \approx w^\top x + b \), and fits the parameters by minimizing squared residuals. It is the canonical baseline for regression because it is interpretable and often has a closed-form OLS solution.
Logistic RegressionLogistic regression is a linear classifier that models the log-odds of a class as \( w^\top x + b \) and maps that score through a sigmoid to get a probability. Despite its name, it is a classification model, not a regression model.
AccuracyAccuracy is the fraction of predictions that are correct, \( (\text{TP}+\text{TN})/N \). It is intuitive and useful when classes are balanced, but it can be badly misleading on imbalanced datasets where always predicting the majority class already yields high accuracy.
PrecisionPrecision is the fraction of predicted positives that are truly positive, \( \text{TP}/(\text{TP}+\text{FP}) \). It matters most when false positives are costly, such as spam filters, safety classifiers, or medical screening follow-ups.
RecallRecall is the fraction of actual positives that the model successfully retrieves, \( \text{TP}/(\text{TP}+\text{FN}) \). It matters most when missing positives is costly, such as fraud detection, disease screening, or retrieval systems where relevant items should not be overlooked.
F1 ScoreF1 score is the harmonic mean of precision and recall, \( 2PR/(P+R) \). It is high only when both precision and recall are high, making it useful for imbalanced classification where accuracy hides the trade-off between false positives and false negatives.
ROUGEROUGE is a family of overlap metrics for summarization and generation, based on matching n-grams, longest common subsequences, or skip-bigrams between a candidate and reference text. It measures lexical recall more than semantic faithfulness, so it is informative but limited.
Longest Common SubsequenceThe longest common subsequence is the longest sequence of symbols that appears in two sequences in the same order, not necessarily contiguously. It underlies edit-distance-style dynamic programming and metrics such as ROUGE-L because it captures shared sequence structure beyond exact n-gram matches.
Edit DistanceEdit distance is the minimum number of insertions, deletions, and substitutions needed to transform one sequence into another. The most common version, Levenshtein distance, is a dynamic-programming measure of string similarity used in spelling correction, alignment, and evaluation.
PerceptronThe perceptron is a linear threshold classifier that predicts a class from the sign of \( w^\top x + b \) and updates its weights only on mistakes. It is historically important because it introduced gradient-like learning for linear separators, but it only converges when the data are linearly separable.
Decision TreeA decision tree predicts by recursively splitting the feature space with if-then tests until a leaf assigns a class or value. Trees are easy to interpret and capture nonlinearity, but a single deep tree has high variance and overfits without pruning or ensembling.
Random ForestA random forest is an ensemble of decision trees trained on bootstrap samples with random feature subsetting at each split. Averaging many decorrelated trees greatly reduces variance, which is why random forests are strong tabular baselines with little tuning.
Support Vector Machine (SVM)A support vector machine finds the decision boundary that maximizes the margin between classes, depending only on the support vectors nearest the boundary. With kernels, SVMs can model nonlinear separators while retaining a convex optimization objective.
Kernel MethodsKernel methods turn linear algorithms into nonlinear ones by replacing inner products with a kernel function that implicitly measures similarity in a higher-dimensional feature space. This is the core trick behind SVMs, kernel ridge regression, and Gaussian processes.
Principal Component Analysis (PCA)Principal component analysis finds orthogonal directions of maximal variance in the data and projects onto the top few of them. It is a linear dimensionality-reduction method that compresses data, denoises features, and reveals dominant global structure through eigenvectors of the covariance matrix.
Dimensionality ReductionDimensionality reduction maps data into fewer dimensions while preserving as much important structure as possible, such as variance, distances, or neighborhood relations. It is used for compression, visualization, denoising, and making downstream learning easier in high-dimensional spaces.
TransformerA Transformer is a sequence model built from self-attention, position-wise MLPs, residual connections, and normalization, rather than recurrence or convolution. Its key advantage is that every token can directly attend to every other token in parallel, which made modern LLM scaling practical.
Decoder BlockA decoder block is the basic unit of a decoder-only Transformer: causal self-attention plus a position-wise MLP, wrapped with residual connections and normalization. Stacking these blocks lets the model mix context across tokens while preserving autoregressive generation.
Decoder-only TransformerA decoder-only Transformer is a Transformer architecture composed only of masked self-attention blocks, so each token can attend only to earlier tokens. This makes it the standard architecture for autoregressive language models such as GPT, LLaMA, and Claude.
Self-AttentionSelf-attention lets each token compute a weighted combination of representations from other tokens in the same sequence, with weights determined by query-key similarity. It is the mechanism that gives Transformers flexible, content-dependent context mixing without recurrence.
Attention ScoreAn attention score is the compatibility value computed between a query and a key before normalization, often by dot product or a learned variant. Higher scores mean the corresponding token or memory slot should receive more weight after the softmax.
What is a scaled attention score?A scaled attention score is a query-key dot product divided by \( \sqrt{d_k} \) before softmax. The scaling keeps the variance of the logits from growing with key dimension, which helps prevent softmax saturation and keeps gradients well behaved.
Masked Attention ScoreA masked attention score is an attention logit after adding a mask that blocks forbidden positions, typically by adding a very large negative value before softmax. This forces the resulting attention weight to be effectively zero at those positions.
Attention WeightsAttention weights are the normalized coefficients, usually produced by a softmax over attention scores, that determine how much each value vector contributes to the output. They form a distribution over positions or memory entries for each query.
Attention MaskAn attention mask is a tensor that tells an attention layer which positions may interact and which must be blocked. It is used for causal generation, padding suppression, and task-specific visibility patterns, and it must be applied before softmax, not after.
Causal MaskA causal mask blocks attention to future positions by masking entries above the sequence diagonal. It enforces left-to-right autoregressive prediction, ensuring that token \( t \) can depend only on tokens \( \le t \).
Multi-Head AttentionMulti-head attention runs several attention mechanisms in parallel on different learned projections of the same input, then concatenates their outputs. This lets the model capture multiple relational patterns at once instead of forcing all interactions through a single attention map.
Attention HeadAn attention head is one parallel query-key-value attention computation inside multi-head attention. Different heads can specialize to different patterns, such as local syntax, long-range dependencies, or induction-like copying behavior.
Grouped-Query Attention (GQA)Grouped-query attention shares key and value heads across groups of query heads, reducing KV-cache size and bandwidth during inference. It sits between full multi-head attention and multi-query attention, preserving most quality while making long-context serving cheaper.
Query, Key, Value (QKV)Query, key, and value are the three learned projections used by attention: the query asks what to look for, the key says what each position offers, and the value is the content returned if that position is attended to. Attention weights come from query-key similarity, but outputs are weighted sums of values.
Projection MatrixA projection matrix is a learned linear map that transforms vectors into another representation space. In Transformers, separate projection matrices create Q, K, and V from hidden states, and another projection maps concatenated head outputs back to the model dimension.
Position-wise MLPA position-wise MLP is the feed-forward sublayer in a Transformer block, applied independently to each token after attention. It adds nonlinearity and channel mixing per token, complementing attention, which mixes information across positions.
Residual Connection (Skip Connection)A residual connection adds a layer's input back to its output, so the layer learns a correction rather than an entirely new representation. This stabilizes optimization, improves gradient flow, and is one reason very deep networks and Transformers train reliably.
Root Mean Square Normalization (RMSNorm)RMSNorm normalizes activations by their root mean square without subtracting the mean. Compared with LayerNorm it is slightly cheaper and often just as effective, which is why many modern LLMs use RMSNorm in place of full mean-and-variance normalization.
Context WindowThe context window is the maximum number of tokens a model can process in one forward pass. It defines the model's accessible working memory at inference time, and longer windows increase both usefulness on long documents and computational cost.
AutoregressionAutoregression is the factorization of a sequence distribution into a product of conditional next-step distributions. In language generation it means producing one token at a time, each conditioned on all previously generated tokens.
PromptA prompt is the text or structured input given to a language model to condition its behavior and output. It can provide instructions, examples, retrieved context, or tool schemas, and in practice it acts as the model's temporary task specification.
System PromptA system prompt is a high-priority instruction block that defines the assistant's role, rules, and behavioral constraints for a conversation. It is usually prepended invisibly to user messages and is intended to override lower-priority prompt content.
Prompting FormatPrompting format is the template used to serialize instructions, roles, examples, and conversation turns into the token sequence a model expects. It matters because the same words in a different format can change model behavior, especially for chat-tuned systems.
Few-Shot PromptingFew-shot prompting includes a small number of labeled examples in the prompt so the model can infer the task from context without updating parameters. It is one of the clearest demonstrations of in-context learning in large language models.
In-Context LearningIn-context learning is the ability of a model to adapt its behavior from instructions or examples placed in the prompt, without changing its weights. The model remains frozen; the adaptation happens within the forward pass through pattern recognition over the context.
Prompt EngineeringPrompt engineering is the practice of designing prompts that make a model reliably produce the desired behavior. It includes choosing instructions, examples, structure, and reasoning scaffolds, and it trades parameter updates for careful interface design.
Chain of ThoughtChain of thought is a prompting strategy that elicits intermediate reasoning steps before the final answer. It often improves performance on multi-step tasks because the model can use the generated text as an external scratchpad rather than compressing all reasoning into one token prediction.
Tree of ThoughtTree of Thought extends chain-of-thought by exploring multiple candidate reasoning paths, evaluating intermediate states, and searching over them with strategies such as BFS or DFS. It is useful when solving the task requires branching, backtracking, or comparing alternative partial plans.
Self-ConsistencySelf-consistency samples multiple reasoning traces for the same problem and chooses the most common final answer rather than trusting a single chain of thought. It often boosts accuracy because different samples make different mistakes, while the correct answer tends to recur.
ReAct (Reason + Act)ReAct is a prompting pattern where a model alternates between reasoning in text and taking actions such as search or tool calls. This lets it use external information and observations to update its plan instead of reasoning only from the original prompt.
Function CallingFunction calling is a language-model capability for producing structured tool invocations instead of only plain text. The model selects a function and arguments that match a schema, which makes tool use more reliable and easier to integrate with software systems.
Large Language Model (LLM)A large language model is a very large neural language model, usually with billions of parameters, pretrained on massive text corpora. Scale gives LLMs broad world knowledge and emergent capabilities such as in-context learning, but the core training objective is still language modeling.
PretrainingPretraining is the large-scale first stage of training where a model learns general-purpose representations from unlabeled or self-supervised data. For LLMs this usually means next-token prediction over massive corpora, producing a base model that later fine-tuning can adapt.
Supervised Fine-Tuning (SFT)Supervised fine-tuning trains a pretrained model on curated input-output pairs so it follows instructions, styles, or task formats more reliably. In chat systems, SFT is the stage that turns a raw completion model into an assistant before preference alignment is applied.
FinetuningFinetuning continues training a pretrained model on a smaller task-specific or domain-specific dataset. It adapts existing representations rather than learning from scratch, which is why it usually needs far less data and compute than pretraining.
Full Fine-TuneA full fine-tune updates all of a model's parameters on the new task or domain. It offers maximum flexibility, but it is much more memory- and compute-intensive than PEFT methods and produces a separate full checkpoint for each adapted model.
Parameter-Efficient Fine-Tuning (PEFT)PEFT is a family of fine-tuning methods that keep most pretrained weights frozen and train only a small number of added or selected parameters. It preserves much of full fine-tuning's quality while reducing memory, compute, and storage costs.
Low-Rank Adaptation (LoRA)LoRA fine-tunes a model by expressing each weight update as a low-rank product \( \Delta W = BA \) while keeping the original weight matrix frozen. This dramatically cuts trainable parameters and optimizer state, which is why LoRA became the default PEFT method for LLMs.
LoRA AdapterA LoRA adapter is the task-specific pair of low-rank matrices inserted around a frozen base weight matrix to produce a learned update at inference or training time. Because adapters are small, many tasks can be stored, swapped, and merged without copying the full base model.
QLoRA (Quantized LoRA)QLoRA combines 4-bit quantization of the frozen base model with LoRA adapters trained in higher precision. This makes fine-tuning very large models feasible on modest hardware because the base weights stay compressed while only the small adapter parameters receive gradient updates.
Base ModelA base model is the pretrained model before instruction tuning, chat alignment, or task-specific fine-tuning. It is usually optimized only for language modeling, so it can complete text well but may not reliably follow user instructions or safety constraints.
Open-Weight ModelAn open-weight model is a model whose trained weights are publicly released for download and local use. That is more specific than 'open source': the weights may be open even when the training data, code, or full recipe are not.
Sampling (in Language Models)Sampling in language models means selecting the next token from the predicted probability distribution instead of always taking the argmax. The decoding rule strongly shapes diversity, coherence, and repetition, which is why temperature, top-k, and top-p matter so much.
Greedy DecodingGreedy decoding always selects the highest-probability next token at each step. It is simple and deterministic, but it often gets trapped in bland or repetitive continuations because it never explores slightly less probable alternatives that might lead to better sequences.
Top-k SamplingTop-k sampling truncates the next-token distribution to the \( k \) most probable tokens, renormalizes, and samples from that set. It removes the low-probability tail that often contains junk while still allowing controlled randomness.
Top-p Sampling (Nucleus Sampling)Top-p, or nucleus, sampling chooses the smallest set of tokens whose cumulative probability exceeds a threshold \( p \), then samples from that adaptive set. Unlike top-k, it expands when the model is uncertain and shrinks when the distribution is sharp.
Frequency PenaltyA frequency penalty subtracts an amount proportional to how many times a token has already appeared, lowering its future logit more with each repetition. It encourages lexical diversity without banning reuse entirely, which makes it gentler than hard repetition constraints.
Presence PenaltyA presence penalty lowers the score of any token that has already appeared, encouraging the model to introduce new words or topics instead of repeating earlier ones. Unlike a frequency penalty, it depends only on whether the token has appeared at least once, not on how many times it appeared.
HallucinationHallucination is when a model produces content that is unsupported or false while presenting it as if it were correct. In language models it often comes from next-token training, weak grounding, or overconfident decoding rather than deliberate deception.
MisalignmentMisalignment is the failure mode where optimizing a model for its training objective or proxy reward does not produce the behavior humans actually want. It includes problems like reward hacking, unsafe shortcuts, and goal pursuit that diverges from the intended specification.
Bias (Fairness)In fairness contexts, bias means systematic differences in treatment or error rates across groups caused by data, labels, measurement, or deployment choices. Fairness asks which notion of equal treatment matters, and different fairness criteria often cannot all be satisfied at once.
ExplainabilityExplainability is the ability to give a human-understandable reason for a model’s prediction or behavior using features, examples, rules, or mechanisms. A good explanation should be useful to a person and, ideally, faithful to what the model actually used.
Retrieval-Augmented Generation (RAG)Retrieval-augmented generation adds a retrieval step so the model conditions on external documents at inference time instead of relying only on memorized parameters. It can improve freshness and grounding, but answer quality depends heavily on retrieval recall, ranking, chunking, and how well the model uses the retrieved evidence.
Semantic SearchSemantic search retrieves results by meaning rather than exact keyword overlap, usually by embedding queries and documents into a vector space and comparing similarity. It handles paraphrases well, but it is often combined with lexical search when exact terms or identifiers matter.
Sparse Mixture-of-Experts (MoE) LayerA sparse mixture-of-experts layer replaces one dense feed-forward block with many expert subnetworks, but routes each token to only a small subset such as top-1 or top-2 experts. This increases parameter count and specialization without increasing per-token compute proportionally.
Router NetworkA router network scores experts or computation paths for each token and decides where that token should be sent in a conditional-compute model such as an MoE. A good router improves specialization while avoiding collapsed routing, overload, and excessive communication.
Expert NetworkAn expert network is one of the specialized submodules inside an MoE layer that processes only the tokens routed to it. Experts usually share the same architecture but learn different functions, so specialization emerges from routing plus load-balancing constraints.
Top-k RoutingTop-k routing sends each token only to the k highest-scoring experts instead of to every expert. This makes MoE computation sparse and efficient, but the choice of k trades off compute cost, robustness, and routing stability.
Load Balancing (MoE)Load balancing in MoE training adds losses or routing constraints so tokens are spread across experts instead of collapsing onto a few popular ones. It matters because uneven routing wastes capacity, creates bottlenecks, and leaves underused experts poorly trained.
Switch TransformerSwitch Transformer is a simplified MoE Transformer that routes each token to exactly one expert in each sparse feed-forward layer. Top-1 routing reduces communication and implementation complexity, enabling very large sparse models, but makes router stability and load balancing especially important.
Model MergingModel merging combines the weights or weight deltas of separately fine-tuned models into one model without full retraining. It can create multitask behavior cheaply, but naive averaging often causes interference unless the merged models share a common base and compatible parameter geometry.
Model SoupsModel soups average the weights of multiple fine-tuned models that lie in the same low-loss basin, often improving accuracy and robustness without extra inference cost. Unlike an ensemble, a soup is still a single model at serving time.
SLERP (Spherical Linear Interpolation)SLERP interpolates between two parameter vectors along the great-circle path on a sphere instead of using straight-line interpolation. In model merging it can preserve vector norms and sometimes produce smoother blends than linear interpolation when the two directions differ strongly.
TIES-MergingTIES-Merging is a model-merging method that reduces interference by trimming small parameter changes, resolving sign conflicts, and merging only updates aligned with an agreed sign. It is designed for cases where separately fine-tuned models disagree on which weights should move and in what direction.
DARE (Drop And REscale)DARE sparsifies fine-tuning deltas by randomly dropping many update entries and rescaling the rest, preserving most behavior while greatly reducing storage and merge interference. It is commonly used as a preprocessing step for delta compression or model merging rather than as a standalone training method.
Task VectorA task vector is the weight difference between a pre-trained model and the same model after fine-tuning on a task. Adding, subtracting, or scaling that vector can steer behavior, so task vectors provide a simple weight-space tool for editing or combining capabilities.
Model CompressionModel compression reduces a model’s memory, latency, or energy cost while trying to preserve performance. Common compression methods include distillation, pruning, quantization, low-rank factorization, and architecture redesign.
Knowledge DistillationKnowledge distillation trains a smaller student model to match the outputs or internal representations of a larger teacher. It transfers some of the teacher’s behavior into a cheaper model, often using soft targets that contain more information than hard labels alone.
PruningPruning removes weights, neurons, heads, or entire blocks that contribute little to performance. It can reduce model size or compute, but aggressive pruning usually needs fine-tuning to recover lost accuracy.
Structured PruningStructured pruning removes whole channels, heads, layers, or blocks, producing regular sparsity that hardware can exploit directly. It usually yields better real-world speedups than unstructured pruning, though it gives less fine-grained control.
Unstructured PruningUnstructured pruning zeros individual weights wherever they appear unimportant, creating irregular sparsity patterns. It can achieve high compression ratios, but specialized kernels are often needed to turn that sparsity into real latency gains.
QuantizationQuantization stores or computes with lower-precision numbers, such as INT8 or 4-bit values, instead of full-precision floats. It reduces memory bandwidth and can speed inference, but accuracy depends on how well the lower-precision representation preserves weights and activations.
Post-Training Quantization (PTQ)Post-training quantization converts a trained model to lower precision after optimization is finished, usually using a calibration set to estimate activation ranges. It is easy and cheap to apply, but accuracy can drop more than with quantization-aware training at very low bit widths.
Quantization-Aware Training (QAT)Quantization-aware training simulates low-precision arithmetic during training so the model learns weights that remain accurate after quantization. It usually outperforms post-training quantization at low bit widths, but it adds training cost and implementation complexity.
Preference-Based AlignmentPreference-based alignment trains models from judgments such as ‘response A is better than response B’ instead of only from supervised targets. It is useful when desired behavior is easier for humans to compare than to specify as a single correct answer.
Reinforcement Learning from Human Feedback (RLHF)RLHF aligns a model by collecting human preference data, training a reward model on those comparisons, and then optimizing the policy to maximize reward while staying close to a reference model. It improved helpfulness and instruction following, but it can also create reward hacking and training instability.
Reward ModelA reward model predicts a scalar preference score for a candidate response, usually from pairwise human comparisons. In RLHF it acts as a learned proxy objective, so the policy can exploit its mistakes if optimization pushes too hard against it.
Self-CritiqueSelf-critique is a prompting or training pattern where a model reviews its own draft, identifies problems, and then revises the answer. It can improve reasoning and safety, but only when the model can recognize errors more reliably than it makes them.
Constitutional AIConstitutional AI aligns a model using an explicit list of principles that guide critique and revision, reducing the need for dense human feedback on every example. The constitution acts like a rule set for self-improvement, though the resulting behavior still depends on the chosen principles and training procedure.
Direct Preference Optimization (DPO)DPO learns directly from preference pairs by making chosen responses more likely than rejected ones without running a separate RL loop. It can be derived from a KL-constrained reward-maximization view, which is why it is often presented as a simpler alternative to PPO-based RLHF.
Bradley-Terry ModelThe Bradley-Terry model turns pairwise comparisons into latent scores by assuming the probability that item A beats item B depends on their score difference. It is widely used for preference modeling, ranking, and reward-model training from pairwise judgments.
Pairwise ComparisonA pairwise comparison asks which of two items is better instead of assigning each item an absolute score. These judgments are often easier and more consistent for humans, which is why they are common in ranking, Elo-style systems, and alignment datasets.
RankingRanking is the task of ordering items by relevance, preference, or utility rather than predicting a single class label. It appears in search, recommendation, and alignment because the main question is which outputs should be placed above others.
Elo RatingElo rating estimates skill from pairwise wins and losses by updating each participant’s score based on expected versus actual outcomes. It was designed for chess, but the same logic is used to aggregate model preferences and benchmark head-to-head evaluations.
Pairwise RankingPairwise ranking learns an ordering from relative preferences between pairs rather than from absolute target values. Many ranking losses optimize the probability that preferred items score above rejected ones, which fits search and alignment data naturally.
Bootstrap ResamplingBootstrap resampling estimates uncertainty by repeatedly sampling with replacement from an observed dataset and recomputing a statistic on each resample. It is useful when analytic uncertainty formulas are hard to derive, though it assumes the sample is reasonably representative.
Confidence IntervalA confidence interval is a range produced by a procedure that would contain the true parameter a fixed fraction of the time over repeated samples, such as 95%. It quantifies estimation uncertainty, but it is not the probability that the parameter lies in this particular realized interval.
Vision Language Model (VLM)A vision-language model jointly processes images and text so it can describe, answer questions about, or reason across both modalities. Most VLMs combine a vision encoder with a language model through projection layers, cross-attention, or joint multimodal pretraining.
Cross-AttentionCross-attention lets one sequence or modality attend to representations produced by another sequence or modality. In encoder-decoder models the decoder queries encoder states, and in multimodal models text tokens often query visual features the same way.
Vision EncoderA vision encoder maps an image into features or tokens that downstream modules can use for classification, retrieval, or generation. CNNs and Vision Transformers are common vision encoders, differing mainly in how they represent spatial structure.
CLIP (Contrastive Language-Image Pre-training)CLIP learns a shared embedding space for images and text by pulling matched image-caption pairs together and pushing mismatched pairs apart. This contrastive objective enables zero-shot classification by comparing an image embedding against text prompts for candidate labels.
Program-Aided Language ModelA program-aided language model uses the LLM to translate a problem into executable code, then lets an interpreter carry out the exact computation. This separates natural-language understanding from symbolic execution and often improves arithmetic or algorithmic reasoning over pure chain-of-thought.
FlashAttentionFlashAttention is an exact attention algorithm that uses tiling and kernel fusion to minimize reads and writes between GPU HBM and on-chip SRAM. It preserves standard attention outputs while greatly reducing memory traffic, which yields large speed and memory gains on long sequences.
Data ParallelismData parallelism replicates the model on multiple devices and splits each batch across them, synchronizing gradients after each step. It is the simplest way to scale training throughput, but every device still stores the full model unless sharding is added.
Model ParallelismModel parallelism splits one model across multiple devices because it is too large or compute-heavy for a single device. The split can happen by layers, tensors, experts, or sequence chunks, trading memory savings for extra communication.
Pipeline ParallelismPipeline parallelism partitions a model by layers across devices and sends microbatches through the partitions like an assembly line. It reduces per-device memory, but pipeline bubbles and stage imbalance can waste throughput if the schedule is poorly tuned.
Tensor ParallelismTensor parallelism shards individual large matrix operations across devices, such as splitting weight matrices by rows or columns. It is effective for very large Transformers, but the frequent collectives mean fast interconnects are important.
Context ParallelismContext parallelism distributes a long sequence across devices so context tokens and their attention-related work are sharded instead of fully replicated. It helps long-context training or inference scale beyond one device, but requires extra communication to preserve exact attention across chunks.
Fully Sharded Data Parallel (FSDP)Fully Sharded Data Parallel shards model parameters, gradients, and optimizer states across data-parallel workers, gathering full parameters only when needed for computation. It is the PyTorch analogue of ZeRO-style training and makes much larger models fit without custom model-parallel code.
Model ShardingModel sharding splits a model’s parameters across devices or storage tiers instead of keeping a full copy everywhere. It is a general systems technique used in tensor parallelism, FSDP, offloading, and large-model serving to reduce per-device memory requirements.
Floating-Point Operations (FLOPs)FLOPs count the number of floating-point arithmetic operations required by a model or workload. They are a useful compute proxy for comparing training or inference cost, though real speed also depends on memory traffic, parallelism, and hardware utilization.
Mixed Precision TrainingMixed-precision training performs most computation in lower precision such as FP16 or bfloat16 while keeping selected quantities in higher precision for stability. It reduces memory use and often increases throughput without much accuracy loss when implemented carefully.
Long-Context PretrainingLong-context pretraining trains or continues training a model on examples with much longer sequences so it learns to use distant context instead of only fitting short windows. It is usually needed because simply changing positional scaling or the context limit does not teach robust long-range retrieval or reasoning.
Needle in a HaystackNeedle in a Haystack is a long-context benchmark that tests whether a model can retrieve a small target fact embedded inside a large distractor context. It is useful for measuring position-sensitive retrieval, but strong needle scores do not guarantee broader long-document reasoning.
Positional EncodingPositional encoding injects token order information into architectures like Transformers whose attention is otherwise permutation-invariant. It can be absolute or relative, and the choice strongly affects extrapolation, long-context behavior, and inductive bias.
Absolute Position EncodingAbsolute position encoding assigns each sequence position its own encoding or embedding and combines it with token representations. It works well inside the trained context range, but it often extrapolates poorly because positions are treated as fixed IDs rather than relative distances.
Relative Position EncodingRelative position encoding represents how far apart tokens are rather than assigning each position a standalone ID. That lets attention depend on distance or offset, which often improves length generalization and transfers patterns more naturally across positions.
Token IDA token ID is the integer index assigned to a token after tokenization. Models do not operate on raw text directly; they look up embeddings from token IDs and later map output logits back to IDs during decoding.
Vocabulary SizeVocabulary size is the number of distinct tokens a tokenizer can emit. A larger vocabulary shortens sequences but increases embedding and softmax size, while a smaller vocabulary produces longer sequences and more token fragmentation.
Subword TokenizationSubword tokenization splits text into frequent pieces smaller than words but larger than individual characters. It handles rare words and open vocabularies well by composing unfamiliar words from known subword units.
Special TokensSpecial tokens are reserved tokens with structural or control meaning, such as BOS, EOS, PAD, SEP, or mask tokens, rather than ordinary text content. They shape formatting, training objectives, and sometimes model behavior.
Padding TokenA padding token is a dummy token added so sequences in a batch have equal length. It should be ignored by the loss and usually masked from attention so it does not behave like real context.
BOS Token (Beginning of Sequence)A BOS token marks the beginning of a sequence and gives the model a consistent start symbol for conditioning generation or encoding. It can help define sequence boundaries and sometimes carries special training semantics.
EOS Token (End of Sequence)An EOS token marks the end of a sequence and tells the model where generation should stop. During training it teaches sequence termination, and during inference it is one of the main stopping conditions.
Attention MechanismAttention computes a context-dependent weighted combination of values, where the weights come from similarities between queries and keys. It lets a model focus on the most relevant parts of an input instead of compressing everything into one fixed vector.
Encoder-Decoder ArchitectureAn encoder-decoder architecture uses an encoder to turn an input sequence into representations and a decoder to generate an output sequence conditioned on those representations. It is the standard design for translation, summarization, and other input-to-output generation tasks.
Sequence-to-Sequence (Seq2Seq)Sequence-to-sequence learning maps one sequence to another, often with different lengths, such as translation or summarization. Modern seq2seq models are usually encoder-decoder Transformers, though earlier versions used recurrent networks with attention.
Distributed RepresentationA distributed representation stores a concept as a pattern across many features or neurons rather than in a single symbolic slot. This supports similarity, composition, and generalization because related concepts can occupy nearby regions of representation space.
Representation LearningRepresentation learning is the process of learning useful features automatically from data rather than hand-engineering them. Good representations preserve the structure that downstream tasks need, such as semantic similarity, invariances, or factors of variation.
Latent SpaceA latent space is the internal feature space in which a model represents inputs after transformation, often in a form that is more compact or task-relevant than raw data. Distances or directions in latent space can encode meaningful variation, but only relative to the model and objective that learned it.
Embedding SpaceAn embedding space is the vector space produced by an embedding model, where tokens, sentences, images, or other objects are mapped to dense numerical representations. Similarity in that space is used for retrieval, clustering, and transfer, though the geometry depends on the training objective.
Nearest Neighbor SearchNearest neighbor search finds the stored vectors most similar to a query under a chosen distance or similarity metric. Exact search is simple but expensive at scale, so large systems often use approximate nearest neighbor indexes instead.
Vector DatabaseA vector database is a system optimized for storing embeddings and retrieving nearest neighbors together with metadata filtering, updates, and persistence. It is the common serving layer behind semantic search and many RAG systems.
BM25BM25 is a sparse retrieval scoring function that ranks documents using term matches weighted by inverse document frequency and document-length normalization. It remains strong for exact lexical search and is often combined with dense retrieval in hybrid systems.
Sparse RetrievalSparse retrieval represents queries and documents with sparse term-based features such as inverted indexes, TF-IDF, or BM25. It excels at exact keywords and rare identifiers, but is weaker than dense retrieval on paraphrases and semantic matching.
Dense RetrievalDense retrieval represents queries and documents with learned dense embeddings and retrieves by vector similarity. It handles paraphrase and semantic matching better than sparse retrieval, but it can miss exact lexical constraints and usually relies on approximate nearest neighbor search.
GroundingGrounding means tying a model’s answer to external evidence, inputs, or world state rather than letting it generate from unsupported priors alone. In RAG or tool-use systems, grounding is what makes outputs traceable to retrieved context or observations.
FactualityFactuality is whether the content of an answer is actually true in the world or according to trusted sources. An answer can be fluent and even faithful to its source while still being nonfactual if the source itself is wrong or outdated.
FaithfulnessFaithfulness is whether a model’s output is supported by the provided input, source document, or chain of evidence. It differs from factuality because a summary can be perfectly faithful to a source that contains false claims.
CalibrationCalibration measures whether predicted probabilities match observed frequencies, so events predicted at 70% should occur about 70% of the time. A model can be accurate but poorly calibrated if its confidence is systematically too high or too low.
Temperature ScalingTemperature scaling calibrates a classifier by dividing logits by a learned scalar temperature before the softmax. It often improves probability calibration on a validation set without changing the model’s ranking of classes.
Batch NormalizationBatch normalization normalizes activations using mini-batch mean and variance, then applies learned scale and shift parameters. It stabilizes optimization and enables deeper networks, but its behavior differs between training and inference because it relies on running statistics.
Layer NormalizationLayer normalization normalizes activations across features within each example rather than across the batch. It works well for variable-length sequences and small batch sizes, which is why it is standard in Transformers.
Gradient ClippingGradient clipping limits gradient norms or values before the optimizer step to prevent unstable updates and exploding gradients. It does not fix a bad objective, but it can stabilize training when rare large gradients would otherwise dominate.
Weight DecayWeight decay shrinks parameters toward zero by multiplying them by a factor slightly below 1 on each optimizer step. In plain SGD it is equivalent to L2 regularization, but in adaptive optimizers the decoupled AdamW form is usually preferred.
Adam OptimizerAdam is an adaptive first-order optimizer that keeps moving averages of the gradient and its square, then bias-corrects them to scale each parameter’s update. It converges quickly and is standard for Transformer training, though it is sensitive to weight decay design and hyperparameters.
AdaGradAdaGrad adapts learning rates by dividing each parameter’s update by the square root of the accumulated historical squared gradients. It works especially well for sparse features, but its learning rates can decay too aggressively over long training runs.
RMSPropRMSProp uses an exponential moving average of squared gradients to normalize updates, preventing the denominator from growing without bound as in AdaGrad. It is useful for nonstationary problems and was a key precursor to Adam.
MomentumMomentum accumulates a running velocity of past gradients so updates keep moving in consistent directions and damp noisy zig-zags. It speeds optimization in ravines and is commonly paired with SGD or Nesterov variants.
Learning Rate ScheduleA learning-rate schedule changes the learning rate over training instead of keeping it constant. Schedules matter because they balance fast early progress with stable late optimization and often determine final performance as much as the base optimizer.
WarmupWarmup starts training with a small learning rate and gradually increases it during the first steps. It reduces early instability, especially in Transformers where large updates before optimizer statistics settle can cause divergence.
Cosine AnnealingCosine annealing decays the learning rate following a cosine curve from a high value to a low value, sometimes with restarts. It provides a smooth schedule that often works well in practice without needing many hand-tuned decay boundaries.
Weight InitializationWeight initialization chooses starting parameter values before training begins. Good initialization keeps activations and gradients in useful ranges so learning can start without vanishing, exploding, or breaking symmetry.
He Initialization (Kaiming Initialization)He initialization sets weight variance to roughly 2/fan-in so ReLU-like activations preserve signal magnitude through depth. It improves on Xavier initialization for one-sided activations that zero out about half the inputs.
Label SmoothingLabel smoothing replaces hard one-hot targets with a mostly-correct probability distribution that assigns a small amount of mass to other classes. It regularizes overconfident classifiers and often improves generalization and calibration, though it can hurt when exact probabilities matter.
Tokenization PipelineA tokenization pipeline is the full process that turns raw text into model-ready inputs, including normalization, pre-tokenization, subword splitting, token-to-ID mapping, truncation, padding, and special-token insertion. Choices here directly affect sequence length, vocabulary coverage, and downstream behavior.
GPU AccelerationGPU acceleration uses highly parallel graphics processors to speed the matrix and tensor operations that dominate modern ML workloads. It matters because deep learning is mostly throughput-bound linear algebra, which GPUs execute far more efficiently than general-purpose CPUs.
CUDACUDA is NVIDIA’s parallel-computing platform and programming model for running general-purpose kernels on GPUs. In machine learning it is the software layer that makes GPU-accelerated training and inference practical, exposing massive parallelism, specialized libraries, and direct control over device memory.
Inference OptimizationInference optimization is the set of techniques that reduce serving latency, memory use, and cost while preserving acceptable quality. Common methods include quantization, batching, KV-cache optimizations, kernel fusion, speculative decoding, and architecture choices that trade a little flexibility for much higher throughput.
Instruction TuningInstruction tuning is supervised fine-tuning on instruction-response examples so a pretrained model learns to follow requests instead of merely continuing text. It improves task generality and usability, but it mainly changes behavior and format-following rather than adding much new world knowledge.
Safety AlignmentSafety alignment is the process of making a model reliably avoid harmful, deceptive, or policy-violating behavior while remaining useful. In practice it combines data curation, supervised tuning, preference optimization or RLHF, classifiers, and adversarial evaluation, but it never guarantees perfect safety.
Red-TeamingRed-teaming is adversarial evaluation in which people or automated systems deliberately try to break a model’s safeguards and expose failure modes. Its purpose is not to improve benchmark scores directly, but to find unsafe or brittle behavior before deployment.
What is a jailbreak in the context of LLMs?In the context of LLMs, a jailbreak is a prompt or interaction pattern that bypasses the model’s safety training or policy enforcement and elicits behavior it was supposed to refuse. Jailbreaks matter because they reveal that aligned behavior can be a thin behavioral layer rather than a deep guarantee.
Adversarial PromptingAdversarial prompting is the deliberate construction of inputs that push a model toward incorrect, unsafe, or unintended behavior. It includes jailbreaks, prompt injection, data exfiltration attempts, and other attacks that exploit weaknesses in instruction-following or context handling.
Benchmark (ML Evaluation)A benchmark in ML evaluation is a standardized task, dataset, metric, and protocol used to compare systems reproducibly. Benchmarks are useful because they make progress measurable, but they can be gamed, saturated, or misaligned with real-world performance.
Human EvaluationHuman evaluation uses people to judge outputs on qualities such as helpfulness, factuality, coherence, or safety that automated metrics often miss. It is usually the most trustworthy evaluation for subjective tasks, but it is expensive, slow, and sensitive to rubric design and annotator variance.
Automated EvaluationAutomated evaluation scores model outputs with metrics or model-based judges instead of human raters. It is fast, scalable, and reproducible, but its usefulness depends on how well the metric correlates with the human judgment that actually matters.
BLEUBLEU is a machine-translation metric based mainly on n-gram precision against one or more reference texts, combined with a brevity penalty. It is useful for corpus-level comparison, but it often misses meaning-preserving paraphrases and is weak as a sentence-level quality measure.
A/B Testing (ML Systems)A/B testing in ML systems is a randomized online experiment that serves different model variants to different user groups and compares outcome metrics. It is the standard way to measure real production impact, because offline wins do not always translate into better user experience.
Continual LearningContinual learning is the problem of learning from a sequence of tasks or data distributions without losing previously acquired capabilities. Its core challenge is the stability-plasticity tradeoff: the model must remain adaptable to new data without catastrophically overwriting old knowledge.
What is catastrophic forgetting?Catastrophic forgetting is the sharp loss of performance on previously learned tasks after a model is trained on new ones. It happens because gradient updates that help the new task can overwrite internal representations that were supporting the old task.
Curriculum LearningCurriculum learning trains a model on examples in an organized order, usually from easier or more structured cases to harder ones. The idea is to improve optimization and generalization by shaping the training distribution, though a bad curriculum can also slow learning or bias the model.
Pretraining CorpusA pretraining corpus is the large unlabeled dataset used to train a model’s base capabilities through self-supervised objectives such as next-token prediction. Its size, quality, duplication rate, domain mix, and filtering choices strongly shape what the model knows and how it behaves.
Instruction DatasetAn instruction dataset is a curated set of prompts paired with preferred responses used to teach a pretrained model how to behave as an assistant. It mainly teaches task framing, format, and interaction style rather than the broad world model that comes from pretraining.
Preference DatasetA preference dataset contains prompts with ranked, paired, or binary-labeled responses indicating which outputs are preferred. It is the standard supervision source for reward modeling and direct preference objectives because it expresses comparative quality better than a single gold answer.
Reward SignalA reward signal is the scalar feedback an RL agent receives about the desirability of its behavior. Because the agent optimizes whatever reward it is given, the design of the reward signal determines whether learning produces the intended behavior or merely exploits a proxy.
Policy (Reinforcement Learning)In reinforcement learning, a policy is the rule that maps states or observations to actions, often as a probability distribution. Learning a policy means directly improving behavior, and in language-model RL the policy is the model’s distribution over tokens or completions conditioned on context.
Value FunctionA value function estimates expected future return, either from a state or from a state-action pair under a policy. It matters because it turns delayed rewards into local training signals, enabling planning, bootstrapping, and lower-variance policy gradients.
AI SafetyAI safety is the broader field concerned with preventing harmful or catastrophic outcomes from advanced AI systems. It includes alignment, robustness, misuse prevention, monitoring, control, and governance, so it is wider than just making a chatbot refuse bad requests.
Alignment (AI)Alignment in AI is the problem of making an AI system’s objectives and behavior match human intentions and values rather than a flawed proxy. The hard part is not only teaching what humans say they want, but ensuring the system pursues that goal robustly in new situations.
InterpretabilityInterpretability is the study of making model behavior understandable to humans, whether by explaining predictions, revealing learned features, or analyzing internal structure. It matters because debugging, trust, scientific understanding, and safety all depend on seeing more than just inputs and outputs.
Mechanistic InterpretabilityMechanistic interpretability treats a neural network as a system to be reverse-engineered into circuits, features, and algorithms. Its goal is not just to correlate neurons with concepts, but to identify the actual internal computations that produce behavior.
Attention VisualizationAttention visualization renders attention weights as heatmaps or token-to-token graphs so we can see which positions a model attends to. It is a useful diagnostic tool, but attention weights alone are not a complete explanation of what the model is computing.
Probing (Neural Networks)Probing tests whether information is encoded in a model’s hidden states by training a simple classifier or regressor on those representations. A successful probe shows that the information is recoverable, but not necessarily that the model causally uses it.
Logit LensLogit Lens maps intermediate hidden states through the final unembedding matrix to inspect what tokens each layer already appears to favor. It is a convenient way to watch a Transformer’s computation unfold, though it is only approximate because earlier layers were not trained to be decoded directly.
Activation AnalysisActivation analysis studies the intermediate activations produced during a forward pass rather than only the model’s static weights. By examining which neurons, channels, or directions fire in different contexts, it helps connect internal representations to model behavior.
Sparse Autoencoder (Mechanistic Interpretability)In mechanistic interpretability, a sparse autoencoder is trained on model activations to decompose dense, superposed representations into a larger set of sparse features. This often makes latent structure more interpretable, because individual learned directions can line up with human-readable concepts or behaviors.
Superposition (Neural Networks)Superposition is the phenomenon in which a network stores more features than it has obvious dimensions by packing them into overlapping directions. It explains why single neurons can look polysemantic and why sparse feature dictionaries are often more informative than neuron-by-neuron inspection.
Foundation ModelA foundation model is a large general-purpose model pretrained on broad data and then adapted to many downstream uses through prompting, fine-tuning, or tool use. Its defining property is transfer: one base model can support many tasks rather than being built for just one.
Scaling LawsScaling laws are empirical relationships showing how loss or capability changes with model size, data, and compute, often following approximate power laws. They matter because they let researchers forecast returns to scale and choose more compute-efficient training regimes.
What are emergent capabilities in large language models?Emergent capabilities in large language models are abilities that look weak or absent at small scale but become strong once the model is large enough. The key caveat is that “emergence” can depend on the metric and threshold used, so apparent jumps are not always literal discontinuities in the underlying capability.
Zero-Shot LearningZero-shot learning is the ability to perform a task from a description alone, without task-specific training examples in the prompt or fine-tuning data. In LLMs it is a direct consequence of broad pretraining and instruction-following ability.
One-Shot LearningOne-shot learning is the ability to learn or generalize from a single labeled example or demonstration. It matters because many real tasks do not provide large datasets, so the model must infer the rule from minimal evidence.
Transfer LearningTransfer learning reuses knowledge learned on one task or dataset to improve performance on another. It is effective because useful features learned in a high-resource setting often remain useful in a lower-resource target domain.
Data AugmentationData augmentation expands a training set with label-preserving transformations such as crops, paraphrases, or noise injection. It improves generalization by teaching the model which variations should not change the answer.
Active LearningActive learning is a training strategy that selectively asks for labels on the most informative unlabeled examples instead of labeling data uniformly at random. Its purpose is to reduce annotation cost by spending human effort where uncertainty or disagreement is highest.
Classification HeadA classification head is the final task-specific layer that maps learned representations to class logits or probabilities. In transfer learning it is often the only part trained from scratch, while the backbone provides reusable features.
Softmax HeadA softmax head is the output projection plus softmax normalization that converts hidden representations into a probability distribution over classes or vocabulary items. In language models it is the layer that turns the final hidden state into next-token probabilities.
Beam SearchBeam search is a decoding algorithm that keeps the top-scoring partial sequences at each step instead of only the single best one. It approximates high-probability generation better than greedy decoding, but it can still miss the global optimum and often reduces diversity.
Repetition PenaltyA repetition penalty is a decoding heuristic that downweights tokens or phrases the model has already used, reducing loops and bland repetition. It improves generation quality when a model is prone to degeneracy, but too much penalty can make text unnatural or incoherent.
Encoder (Transformer)A Transformer encoder is a stack of self-attention and feed-forward blocks that builds contextual representations of an input sequence. Because encoder self-attention is usually bidirectional, it is well suited for understanding tasks such as classification, retrieval, and sequence labeling.
Decoder (Transformer)A Transformer decoder is the autoregressive half of the architecture that predicts tokens using causal self-attention and, in encoder-decoder models, optional cross-attention to an encoder output. Its defining constraint is that each position can attend only to earlier positions when generating.
Neural Language ModelA neural language model predicts text with learned distributed representations and a neural network rather than count tables. Its main advantage over classical n-gram models is that it can generalize to unseen contexts by sharing statistical strength across similar words and patterns.
Semantic SpaceA semantic space is an embedding space in which geometric relations reflect meaning, similarity, or functional role. Nearby points tend to correspond to semantically related items, which is why vector search and representation learning work at all.
Information RetrievalInformation retrieval is the problem of finding and ranking the documents, passages, or items most relevant to a query. Modern systems combine lexical matching, learned embeddings, and ranking models because exact term overlap and semantic similarity each capture different kinds of relevance.
Logit AdjustmentLogit adjustment means modifying logits to account for effects such as class imbalance, prior shift, or calibration goals before taking probabilities or losses. It changes the decision boundary in a simple way by shifting scores rather than changing the underlying representation.
Softmax NormalizationSoftmax normalization converts a vector of logits into a probability distribution by exponentiating each score and dividing by the total. It preserves rank order while making outputs comparable, which is why it is the standard final normalization for multiclass prediction.
Over-ParameterizationOver-parameterization means a model has far more parameters than the minimal number apparently needed to fit the data. Counterintuitively, this often helps optimization and can still generalize well because training dynamics and implicit regularization matter as much as raw parameter count.
Domain-Specific PretrainingDomain-specific pretraining continues or repeats pretraining on corpus data from a specialized domain such as law, medicine, or code. It improves vocabulary use, factual recall, and style in that domain, but it can also narrow the model or erode performance outside the target distribution.
Distributed Computing (ML Training)Distributed computing in ML training spreads computation, memory, or both across many devices and often many machines. It is what makes modern large-model training possible through strategies such as data parallelism, model parallelism, sharding, and pipeline execution.
Memory Optimization (ML Training)Memory optimization in ML training is the collection of techniques that reduce peak memory so larger models or batches fit on available hardware. Common examples are mixed precision, activation checkpointing, optimizer sharding, offloading, and more memory-efficient attention kernels.
Conversational AIConversational AI is a class of systems designed for multi-turn interaction, where the model must respond helpfully while tracking context, intent, and dialogue state. The hard part is not generating one good answer, but remaining coherent and useful across an extended interaction.
Role-Playing (LLMs)Role-playing in LLMs means conditioning a model to adopt a persona, voice, or behavioral frame during generation. It is useful for simulation and product design, but it also shows how easily high-level behavior can be steered by prompt context.
Model EvaluationModel evaluation is the systematic measurement of how a model performs, fails, and trades off across tasks, metrics, and deployment contexts. Good evaluation combines offline benchmarks, stress tests, human judgment, and online metrics rather than relying on a single score.
Shadow DeploymentShadow deployment runs a new model in production alongside the live system without letting its outputs affect users. This makes it possible to compare latency, quality, and failure modes on real traffic before committing to a risky rollout.
Feedback Loop (ML Systems)A feedback loop in an ML system occurs when the model’s outputs change the data it will later train on or be evaluated against. These loops can reinforce bias, distort demand, and make offline metrics look better even while the real system gets worse.
Data MixtureA data mixture is the weighting and composition of different datasets or domains in a training run. It matters because capability, robustness, and bias often depend as much on what proportion of the data comes from each source as on the total token count.
GRPO (Group Relative Policy Optimization)GRPO is a policy-optimization method that scores sampled responses relative to others in the same group, using those relative rewards to update the policy. Its appeal is that it can improve reasoning performance while avoiding some of the memory overhead of PPO-style critic training.
Bias MitigationBias mitigation is the set of methods used to reduce unfair or systematically skewed behavior in models and datasets. It can act before training, during optimization, or after deployment, but every intervention trades off fairness goals, accuracy, and operational complexity.
Transparency (AI Systems)Transparency in AI systems means making system behavior, limitations, provenance, and decision pathways inspectable to users, developers, or regulators. It is broader than interpretability because it includes documentation, reporting, and operational visibility, not just internal model analysis.
Steering VectorsSteering vectors are directions in activation space that, when added to hidden states, systematically change model behavior toward traits such as refusal, sentiment, or persona. They are useful because they show that some behaviors can be modified directly in representation space without full retraining.
Activation PatchingActivation patching is a causal analysis method where activations from one run are inserted into another to test which components matter for a given behavior. If patching a layer or head restores the behavior, that component is evidence for being on the relevant causal path.
Log-Sum-Exp TrickThe log-sum-exp trick computes expressions like log(sum(exp(x_i))) stably by subtracting the maximum logit before exponentiation. It prevents overflow and underflow, so it is a standard numerical tool in softmax, cross-entropy, and probabilistic inference.
Exponential Family of DistributionsThe exponential family is the class of distributions that can be written in the form exp(eta^T T(x) - A(eta) + c(x)). This shared form gives them sufficient statistics, convenient conjugate priors, and clean maximum-likelihood geometry, which is why they dominate classical statistical modeling.
Kolmogorov-Arnold NetworksKolmogorov-Arnold Networks replace fixed scalar weights on edges with learnable one-dimensional functions, so layers are built from sums of learned univariate transforms rather than simple affine maps. They are motivated by the Kolmogorov-Arnold representation theorem and are often discussed as a more interpretable alternative to MLPs, not a universal replacement.
KL DivergenceKL divergence measures how one probability distribution differs from a reference distribution through an expected log-ratio. It is nonnegative and asymmetric, so it is best understood not as a distance but as the penalty for modeling samples from one distribution with another.
Jensen's InequalityJensen’s inequality says that for a convex function f, applying f after taking an expectation gives a value no larger than taking the expectation after applying f. This one fact underlies many core results in ML, including the nonnegativity of KL divergence and the derivation of variational lower bounds.
Bayes' TheoremBayes’ theorem updates beliefs by combining a prior with the likelihood of observed evidence to produce a posterior. In compact form, posterior is proportional to likelihood times prior, which is why Bayesian inference is fundamentally a rule for disciplined belief revision.
Expectation–Maximization (EM) AlgorithmThe EM algorithm is an iterative method for maximum-likelihood or MAP estimation in models with latent variables. Each round first estimates the hidden structure under the current parameters and then re-optimizes the parameters as if that hidden structure were known.
Eigenvalues and EigenvectorsFor a matrix A, an eigenvector is a nonzero direction that A only rescales, and the scaling factor is its eigenvalue. Eigenpairs matter because they reveal invariant directions, control stability, and make problems like PCA and spectral clustering possible.
Jacobian and HessianThe Jacobian collects first-order partial derivatives of a vector-valued function, while the Hessian collects second-order partial derivatives of a scalar function. Together they describe local sensitivity and curvature, which is why they are central to optimization and dynamical analysis.
Multivariate Gaussian DistributionThe multivariate Gaussian is the vector-valued generalization of the normal distribution, parameterized by a mean vector and covariance matrix. It is foundational because linear transformations, marginals, and conditionals all stay Gaussian, making analysis and inference unusually tractable.
Naive Bayes ClassifierNaive Bayes is a probabilistic classifier that applies Bayes’ theorem under the simplifying assumption that features are conditionally independent given the class. That assumption is usually false, but the model is still fast, data-efficient, and surprisingly effective for sparse text problems.
k-Nearest Neighbors (k-NN)k-nearest neighbors predicts by finding the k closest labeled training points to a query and then voting or averaging their labels. It has almost no training phase, but its predictions depend heavily on the distance metric and it degrades in high-dimensional spaces.
Universal Approximation TheoremThe universal approximation theorem says that a sufficiently wide neural network with a suitable nonlinearity can approximate any continuous function on a compact domain arbitrarily well. It is an existence result, not a guarantee that training will find that approximation efficiently.
Double DescentDouble descent is the phenomenon in which test error first follows the classical U-shape with increasing model size, then improves again once the model passes the interpolation threshold. It matters because it shows that the old bias-variance story is incomplete in highly overparameterized regimes.
GrokkingGrokking is a delayed generalization phenomenon in which a model first memorizes the training set and only much later snaps into a simple algorithm that generalizes well. It is interesting because the model already had enough capacity to fit the data, yet the more general solution emerged only after long training and regularization pressure.
Neural Tangent Kernel (NTK)The Neural Tangent Kernel is the kernel that describes how an infinitely wide network trained by small gradient steps evolves around its initialization. In that limit, training becomes equivalent to kernel regression, which explains part of the behavior of very wide networks.
State Space Models / MambaState space models such as Mamba process sequences by evolving a learned hidden state through recurrence rather than full quadratic attention. Their main appeal is linear-time sequence processing with strong long-context efficiency, especially when selective state updates let the model decide what to remember.
Linear AttentionLinear attention is the family of attention mechanisms that rewrites or approximates softmax attention so sequence processing scales roughly linearly instead of quadratically with length. The benefit is efficiency on long contexts, but the tradeoff is that exact softmax behavior is usually lost.
ALiBi (Attention with Linear Biases)ALiBi is a positional method that adds head-specific linear distance penalties directly to attention logits instead of injecting separate position embeddings. Because the bias is built into the score function, models trained with ALiBi often extrapolate to longer contexts better than models tied to a fixed embedding table.
YaRN / NTK-aware RoPE ScalingYaRN and other NTK-aware RoPE-scaling methods extend the usable context of RoPE-based models by rescaling or interpolating rotary frequencies rather than retraining the model from scratch. Their goal is to preserve short-context behavior while making long-range positions less distorted.
Sliding-Window AttentionSliding-window attention restricts each token to attending only within a local context window rather than the entire sequence. This reduces compute and memory from full-context attention and is effective when most useful dependencies are nearby, though it can miss long-range interactions unless combined with global mechanisms.
Multi-head Latent Attention (MLA)Multi-head Latent Attention compresses the keys and values of multi-head attention into a smaller latent representation before use. Its main advantage is a much smaller KV cache and lower decode-time memory bandwidth, which is why it is attractive for long-context serving.
Chinchilla Scaling LawsChinchilla scaling laws showed that many large language models were undertrained for their size under fixed compute budgets. The central prescription is to train smaller models on more tokens than the earlier parameter-heavy frontier, yielding better compute-optimal performance.
Auxiliary Load-Balancing Loss (MoE)The auxiliary load-balancing loss in a Mixture-of-Experts model encourages the router to spread tokens more evenly across experts. Without it, routing often collapses onto a few experts, which wastes capacity and creates severe hot spots in both learning and systems performance.
Muon OptimizerMuon is an optimizer designed especially for matrix-valued parameters that replaces the raw update direction with an orthogonalized one. The point is to respect matrix structure rather than treating every weight tensor as a flattened vector, with the goal of improving training efficiency relative to standard first-order optimizers.
Pretraining Data DeduplicationPretraining data deduplication removes near-duplicate documents or passages from a training corpus. It improves per-token efficiency and reduces memorization and benchmark contamination, because repeatedly seeing the same text usually wastes compute more than it adds knowledge.
REINFORCE AlgorithmREINFORCE is the basic Monte Carlo policy-gradient algorithm that updates parameters by weighting the log-probability of sampled actions by their returns. It is unbiased, but its variance is high, which is why practical methods usually add baselines or critics.
Generalized Advantage Estimation (GAE)Generalized Advantage Estimation is a family of advantage estimators that interpolates between low-variance temporal-difference updates and high-variance Monte Carlo returns using a parameter lambda. It is widely used because it gives a practical bias-variance tradeoff for policy-gradient training.
Actor–Critic MethodsActor-critic methods learn a policy and a value estimator at the same time. The actor chooses actions, while the critic estimates return and supplies a lower-variance learning signal than raw returns alone.
Process Reward Models (PRM) vs Outcome Reward Models (ORM)Outcome reward models score only the final answer, while process reward models score the intermediate steps of a solution. PRMs provide denser supervision and better guidance for search and long-form reasoning, but they require more fine-grained labels and more complex evaluation.
KTO (Kahneman–Tversky Optimization)KTO is a preference-optimization objective that learns from binary desirable-versus-undesirable labels instead of pairwise rankings. It uses a utility formulation inspired by prospect theory, making it a cheaper alternative when collecting full preference comparisons is too expensive.
RLAIF (RL from AI Feedback)RLAIF replaces human preference labels with judgments produced by another AI model following a rubric. It scales alignment data collection much more cheaply than RLHF, but it also transfers the biases and blind spots of the judge model into the training signal.
Reasoning Models (o1 / R1-style Long-CoT)Reasoning models in the o1 or R1 style are language models trained or prompted to spend extra inference compute on long multi-step reasoning before answering. Their key idea is that better reasoning can come not only from bigger models, but from better search, verification, and credit assignment at inference and post-training time.
Process SupervisionProcess supervision trains a model on the quality of intermediate reasoning steps rather than only on whether the final answer is correct. It improves credit assignment for long solutions and makes verification more local, though collecting reliable step-level labels is expensive.
Monte Carlo Tree Search for LLM ReasoningMonte Carlo Tree Search for LLM reasoning treats partial solution paths as tree nodes, expands candidate continuations, and uses rollouts or value estimates to decide where to search next. It is attractive because it turns one-shot generation into guided search over reasoning trajectories instead of committing immediately to a single chain of thought.
Self-Refine / ReflexionTwo closely related inference-time techniques in which an LLM critiques and revises its own output over multiple rounds. Self-Refine uses the same model in three roles (generate → feedback → refine); Reflexion adds an episodic memory of past failures to guide future trajectories in agentic tasks.
vLLM & Continuous BatchingvLLM is an LLM serving system built around PagedAttention and continuous batching. Instead of waiting for a batch to finish, it admits and schedules requests at each decoding step, which reduces padding waste and improves throughput for variable-length generations.
GPTQ QuantizationA one-shot, layer-wise post-training quantization that pushes LLM weights to 3–4 bits while preserving generation quality. GPTQ reformulates quantization as a per-row error-minimisation problem solved greedily using the inverse Hessian of a small calibration set, achieving 3-bit LLaMA-65B with <1% perplexity loss.
AWQ (Activation-aware Weight Quantization)AWQ is a post-training quantization method that preserves quality by protecting the weights attached to the largest activation channels before rounding. It targets low-bit LLM inference with small accuracy loss and is popular because it is simpler to deploy than Hessian-based methods such as GPTQ.
Attention Sinks / StreamingLLMAttention sinks are the first few tokens in a causal Transformer that absorb disproportionate attention from later positions, even when they carry little semantic content. StreamingLLM exploits this by keeping sink tokens and a short recent window in the KV cache, enabling long streaming inference with bounded memory.
Induction HeadsA specific two-head circuit in Transformer attention that copies the next token after a previous occurrence of the current token — the computational basis for in-context learning. Anthropic showed induction heads form suddenly during training, coinciding with the sharp jump in ICL ability.
Circuit AnalysisThe mechanistic-interpretability practice of identifying subgraphs of weights, residual-stream components, and attention heads that jointly implement a human-interpretable algorithm (indirect object identification, modular addition, greater-than). Circuit analysis produces falsifiable, causal accounts of what a network has learned.
ROME / MEMIT Model EditingRank-one edits to MLP weights that inject a single fact (ROME) or thousands of facts (MEMIT) into a pretrained LLM without retraining. They exploit the observation that MLP blocks act as key–value memories, identify the causal neurons via activation patching, and solve a closed-form optimisation problem for the minimal-norm weight update.
Vision Transformer (ViT)Dosovitskiy et al. (2020) showed that a pure Transformer applied to fixed-size image patches as tokens matches or exceeds state-of-the-art CNNs on ImageNet when pretrained on enough data. ViT is the backbone of modern vision-language models (CLIP, SigLIP, DINOv2, MAE) and the foundation of nearly all 2020s visual representation work.
Masked Autoencoder (MAE)A self-supervised ViT pretraining objective: randomly mask 75% of image patches and train an asymmetric encoder–decoder to reconstruct pixel values from the visible 25%. MAE is simple, compute-efficient (the encoder sees only unmasked patches), and produces state-of-the-art ImageNet fine-tuning representations.
DINOv2A self-supervised ViT pretraining recipe from Meta (Oquab et al., 2023) that combines a DINO-style self-distillation objective with an iBOT masked-patch prediction objective and a curated 142M-image dataset. DINOv2 produces general-purpose frozen visual features that outperform task-specific supervised baselines on classification, segmentation, depth, and correspondence.
SigLIPA contrastive image–text pretraining method (Zhai et al., 2023) that replaces CLIP's softmax-over-batch contrastive loss with a pairwise sigmoid binary cross-entropy. SigLIP removes the need for large global batches, scales batch-size-efficiently, and achieves CLIP-level or better zero-shot accuracy at a fraction of the training compute.
Denoising Diffusion Probabilistic Models (DDPM)A generative model that learns to reverse a fixed Gaussian corruption process. Ho et al. (2020) showed that predicting the added noise with a neural network, trained by a simple MSE loss on \( T \) diffusion steps, yields state-of-the-art image synthesis — the foundation of all modern image/video diffusion.
Classifier-Free GuidanceClassifier-free guidance is a sampling trick for conditional diffusion models that combines conditional and unconditional predictions to push samples harder toward the prompt. It improves prompt adherence without a separate classifier, but too much guidance can oversaturate images and reduce diversity.
Flow Matching / Rectified FlowA generative modelling framework that learns a time-dependent velocity field mapping noise to data along a fixed probability path. Rectified Flow in particular learns straight-line paths between noise and data samples, enabling 1–4 step sampling with quality matching much deeper diffusion models.
Latent DiffusionRun the diffusion process in the compressed latent space of a pretrained VAE rather than in pixel space. Latent diffusion (Rombach et al., 2022) slashes memory and compute by ~8× for images while preserving sample quality, and is the architecture behind Stable Diffusion, SDXL, SD3, and most text-to-image systems.
Core LLM Benchmarks (MMLU, HumanEval, GSM8K, MATH)MMLU tests broad academic knowledge, HumanEval tests code generation by unit tests, GSM8K tests grade-school math word problems, and MATH tests harder symbolic reasoning. Together they cover knowledge, code, and reasoning, but all can be gamed or saturated, so they are only a partial view of model quality.
LMSYS Chatbot ArenaLMSYS Chatbot Arena is a crowdsourced pairwise-evaluation platform where users compare two anonymous models by chatting and voting. Its Elo-style ranking captures interactive preference better than a single benchmark, but it is noisy and sensitive to traffic mix and prompt selection.
Chain Rule of ProbabilityThe factorisation \( p(x_1, \dots, x_n) = \prod_{i=1}^n p(x_i \mid x_{<i}) \) that decomposes any joint distribution into a product of conditionals. It is the mathematical bedrock of autoregressive language models, belief networks, and most tractable density estimation.
Central Limit TheoremGiven i.i.d. samples \( X_1, \dots, X_n \) with finite mean \( \mu \) and variance \( \sigma^2 \), the standardised sample mean \( \sqrt{n}(\bar X_n - \mu)/\sigma \) converges in distribution to \( \mathcal{N}(0,1) \). The CLT underlies confidence intervals, stochastic-gradient noise analysis, and many initialisation arguments.
Variance, Covariance and CorrelationVariance measures the spread of a single random variable; covariance measures joint variation of two; correlation normalises covariance to \( [-1, 1] \) and is scale-free. Together they form the second-order statistics that drive PCA, linear regression, Kalman filters, and most initialisation schemes.
Taylor Series ExpansionApproximates a smooth function near a point \( a \) by a polynomial whose coefficients are the function's derivatives: \( f(x) \approx \sum_{k=0}^{K} \frac{f^{(k)}(a)}{k!}(x-a)^k \). Provides the theoretical scaffolding for gradient descent (1st order), Newton's method (2nd order), Laplace approximations, and loss-landscape analysis.
Entropy vs Cross-Entropy vs KL (Unified View)A single identity \( H(p,q) = H(p) + D_{\text{KL}}(p \| q) \) ties the three together: entropy is the optimal code length under \( p \); cross-entropy is the code length using \( q \); KL is the extra cost of the mismatch. This view clarifies why minimising classification cross-entropy is equivalent to MLE and to minimising KL.
Mahalanobis DistanceThe metric \( d_M(\mathbf{x}, \boldsymbol{\mu}) = \sqrt{(\mathbf{x}-\boldsymbol{\mu})^\top \Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})} \) measures distance in a space whitened by the covariance \( \Sigma \). It is the natural distance for Gaussian data, the log-likelihood core of a multivariate Gaussian, and the basis of Fisher's discriminant analysis.
Conjugate PriorsA prior is conjugate to a likelihood family if the posterior belongs to the same family as the prior. Conjugacy turns Bayesian updating into a closed-form parameter update and underlies analytical treatments of Beta–Binomial, Dirichlet–Multinomial, and Normal–Normal models.
Gaussian Mixture Models (GMM)A density model \( p(\mathbf{x}) = \sum_{k=1}^{K} \pi_k \, \mathcal{N}(\mathbf{x}; \boldsymbol{\mu}_k, \Sigma_k) \) that represents data as a weighted sum of Gaussian components. Fitted by the EM algorithm, GMMs are the canonical example of latent-variable density estimation and the statistical cousin of \( k \)-means.
Variational Inference / ELBOA framework that turns Bayesian inference into optimisation: choose a tractable family \( q(\mathbf{z}; \phi) \) and maximise the evidence lower bound \( \mathcal{L}(\phi) = \mathbb{E}_q[\log p(\mathbf{x},\mathbf{z})] - \mathbb{E}_q[\log q(\mathbf{z})] \), which simultaneously approximates the posterior and bounds the log-evidence. VAEs, variational Bayes, and amortised inference all descend from this objective.
Evidence Lower Bound (ELBO) — DerivationTwo derivations of the ELBO: (i) Jensen's inequality applied to \( \log p(\mathbf{x}) = \log \int p(\mathbf{x}, \mathbf{z})\,d\mathbf{z} \), and (ii) the identity \( \log p(\mathbf{x}) = \text{ELBO}(q) + D_{\text{KL}}(q \| p(\cdot \mid \mathbf{x})) \). Both produce the same bound, but the second makes the gap explicit.
Score MatchingAn estimation principle (Hyvärinen, 2005) that fits an unnormalised density by matching the model's score \( \nabla_{\mathbf{x}} \log p_\theta(\mathbf{x}) \) to the data's score. Integration-by-parts eliminates the unknown data-score, yielding a tractable objective that underlies modern score-based diffusion models.
Denoising Score MatchingVincent (2011) showed that the score of a Gaussian-corrupted data distribution \( q_\sigma(\tilde{\mathbf{x}} \mid \mathbf{x}) \) admits a closed-form target, reducing score learning to a simple regression: predict \( (\mathbf{x} - \tilde{\mathbf{x}})/\sigma^2 \). This identity is the algorithmic heart of modern diffusion models.
DDIM (Deterministic Diffusion Sampling)Song, Meng & Ermon (2020) introduced a non-Markovian sampler for DDPM-trained diffusion models that generates samples in a fraction of the steps while matching quality. DDIM's \( \eta = 0 \) limit is a deterministic ODE integrator, enabling latent interpolation and invertibility.
Consistency ModelsSong et al. (2023) train a neural network \( f_\theta(\mathbf{x}_t, t) \) whose output is consistent along the probability-flow ODE trajectories of a diffusion model, so that \( f_\theta(\mathbf{x}_t, t) \approx \mathbf{x}_0 \) for every \( t \). This collapses diffusion sampling to a single step, with optional multi-step refinement for quality.
Diffusion Transformers (DiT)Peebles & Xie (2022) replace the U-Net backbone of latent diffusion with a standard Transformer over VAE-latent patches. DiT scales predictably with compute, matches or exceeds U-Net quality, and is the architectural backbone of Stable Diffusion 3, Sora, and most frontier text-to-image/video diffusion models.
Stable Diffusion PipelineA text-to-image pipeline composed of (i) a VAE that compresses pixels to a 64×-smaller latent, (ii) a text encoder (CLIP) that provides conditioning, and (iii) a diffusion U-Net (or DiT) that denoises in latent space. All three pretrained components are glued by classifier-free guidance at inference.
Contrastive Learning (SimCLR / MoCo)Self-supervised visual representation learning via the InfoNCE loss: pull together two augmented views of the same image (positives) while pushing apart views of all other images (negatives). SimCLR uses in-batch negatives; MoCo uses a queued momentum encoder, enabling large effective negative pools with small batches.
BYOL / Self-DistillationBootstrap Your Own Latent (Grill et al., 2020) shows that strong visual representations can be learned without negatives : an online network predicts the output of a momentum-updated target network on a different augmentation of the same image. The same template underlies DINO, MoCo-v3, and other non-contrastive SSL methods.
JEPA (Joint Embedding Predictive Architecture)LeCun's self-supervised template (2022) that predicts the representation of a target from the representation of a context, rather than predicting the target itself. By regressing in embedding space, JEPA avoids wasting capacity on irrelevant per-pixel detail; I-JEPA and V-JEPA are concrete instantiations for images and video.
Vision-Language Contrastive Objectives Beyond CLIPSuccessors to CLIP refine its symmetric InfoNCE loss for better efficiency, finer-grained alignment, and scaling. SigLIP replaces softmax with a pairwise sigmoid; LiT freezes a pretrained image tower; ALIGN scales noisily; FILIP does token-level contrast; CoCa adds a captioning head. All share the joint-embedding template.
World ModelA learned predictive model of environment dynamics — given a state and action, predict the next state (and reward). World models enable planning, offline rollouts, and sample-efficient RL. Recent systems (Dreamer, MuZero, Genie) show that powerful latent world models scale to games, robotics, and interactive video generation.
Whisper (Speech-to-Text)OpenAI's 2022 encoder-decoder Transformer trained on 680k hours of weakly supervised multilingual audio-text pairs. Whisper performs speech recognition, translation, and voice-activity / language ID from a single model, with strong zero-shot robustness to noise, accent, and domain shift.
Encodec / Neural Audio CodecsMeta's Encodec (2022) is a neural audio codec that compresses audio to discrete tokens via residual vector quantisation (RVQ) and reconstructs it with a neural decoder. Encodec is the tokeniser of choice for generative audio models (AudioLM, MusicGen, VALL-E), bridging continuous audio and LLM-style discrete modelling.
Residual Networks (ResNet as Architecture)A residual network replaces a plain layer stack with blocks that learn a residual update \(F(x)\) and add it back to the input, so each block computes \(y = x + F(x)\). This makes very deep CNNs much easier to optimize and triggered the shift from VGG-style stacks to residual architectures.
Transformer-XL / Segment-Level RecurrenceDai et al. (2019) extend Transformers beyond fixed context by caching hidden states of the previous segment and allowing attention to read from them — a simple "segment-level recurrence" that gives an effective receptive field of \( N \cdot L \) for \( L \) layers and segment length \( N \). Paired with relative positional encoding, it was a key bridge between pure attention and long-context models.
Longformer / BigBird (Sparse Long-Context Attention)Fixed sparsity patterns that reduce attention from \( O(n^2) \) to \( O(n) \) for long documents. Longformer combines sliding-window + global attention; BigBird adds random attention and proves the result retains full-attention universal-approximation properties. Both were pre-2022 answers to scaling Transformers to 4k–16k tokens.
RetNet / Retention NetworksSun et al. (2023) introduce a Transformer-alternative block whose retention operator admits three equivalent forms: parallel (for training), recurrent (for \( O(1) \) inference per token), and chunkwise-recurrent (for long-sequence training). RetNet aims for RNN-like inference cost with Transformer-like parallelisable training.
RWKVAn RNN-Transformer hybrid (Peng et al., 2023) whose block is a parallelisable linear-attention operation at training time and a simple recurrent state update at inference time. RWKV scales to 14B+ parameters with Transformer-competitive perplexity, offering constant-memory inference.
Hyena / Long ConvolutionsPoli et al. (2023) propose replacing attention with a data-controlled long-range convolution: a filter parameterised implicitly by an MLP-of-positions, applied via FFT for \( O(n \log n) \) cost. Hyena approaches Transformer quality on pretraining perplexity at a fraction of the compute.
xFormers / Memory-Efficient AttentionA library / pattern of attention implementations that avoid materialising the \( n \times n \) attention matrix, reducing memory from \( O(n^2) \) to \( O(n) \). xFormers bundles FlashAttention, Memory-Efficient Attention (Rabe & Staats), block-sparse variants, and ALiBi/RoPE patches under a unified API — a precursor to the default attention kernels shipped in PyTorch 2.0+.
KV Cache Compression (H2O, SnapKV)Inference-time methods that shrink a long-context KV cache by evicting tokens that contribute little to future attention. H2O (Zhang et al., 2023) evicts by cumulative attention score; SnapKV (Li et al., 2024) observes that recent queries already reveal which past tokens matter, enabling one-shot pre-fill-time compression.
Chunked PrefillA serving-time technique that breaks the long prefill of a prompt into small chunks and interleaves them with decode steps of other requests. By keeping GPU utilisation high during prefill and avoiding long tail latencies, chunked prefill dramatically improves throughput in mixed-batch LLM serving.
Disaggregated Prefill/Decode ServingDisaggregated prefill/decode serving splits prompt processing and token-by-token decoding onto different GPU pools and transfers the KV cache between them. This reduces contention because prefill is throughput-heavy while decode is latency-sensitive, improving utilization in large serving clusters.
Paged vs Block KV CacheTwo allocation strategies for an LLM's growing KV cache. Block (contiguous) allocation pre-reserves the worst-case length per request and wastes memory. Paged (PagedAttention, vLLM 2023) allocates fixed-size pages on demand and chains them like OS virtual memory, yielding 2–4× higher batch-size at the cost of kernel-level bookkeeping.
Tensor Cores & GEMM FundamentalsTensor cores are specialised matrix-multiply units in NVIDIA GPUs (introduced in Volta, 2017) that execute small mixed-precision matrix-multiply-accumulate (MMA) ops per clock. Peak DL throughput is set by tensor-core flops; structured matrix multiplies (GEMM) that tile the problem to tensor-core shape are how deep learning touches that ceiling.
BF16 / FP8 / MXFP4 Number FormatsThese are low-precision number formats used to trade numerical precision for speed and memory efficiency in modern ML hardware. BF16 is the standard training workhorse, FP8 is increasingly used for faster training and inference, and 4-bit floating formats push efficiency further for aggressive inference optimization.
DeepSpeed ZeRO-Infinity / OffloadingAn extension of ZeRO that offloads optimizer states, gradients, and parameters to CPU RAM and NVMe SSDs, enabling training of trillion-parameter models on modest GPU clusters. ZeRO-Infinity (Rajbhandari et al., 2021) uses bandwidth-aware partitioning and overlap to hide the offload latency.
Ring Attention / Context Parallel for Long SequencesA distributed-attention algorithm that shards an \( n \)-token sequence across \( P \) devices and computes each attention output via a ring of key-value rotations. Ring Attention (Liu et al., 2023) enables context lengths of millions of tokens on multi-GPU clusters with near-linear scaling.
Rejection Sampling Fine-Tuning (RFT)An offline alignment method: sample many completions from a base model, keep only those that pass a quality / reward filter, and fine-tune on the survivors. RFT is the simplest way to convert a reward model (or a verifier) into improved policy behaviour, and serves as a strong baseline against DPO / PPO.
Self-Play Fine-Tuning (SPIN)Chen et al. (2024) cast fine-tuning as a game between the current model and its previous iterate: the new model must distinguish its own generations from demonstrations, improving it without any new human data. SPIN delivers substantial improvements on supervised data alone, bootstrapping from a weak SFT model toward a stronger one.
Iterative DPOIterative DPO repeats a simple loop: sample responses from the current policy, score or label them, apply a DPO-style update, and repeat. It brings online data collection to preference optimization while keeping DPO's simpler training dynamics compared with PPO-based RLHF.
Best-of-N Sampling and its ScalingAn inference-time boost: sample \( N \) responses from a language model, score them with a reward model or verifier, and return the best. Quality scales with \( \log N \); scaling laws predict the knob where best-of-\( N \) inference compute equals extra training compute, and motivate distillation of best-of-\( N \) behaviour back into the policy via RFT.
Weak-to-Strong GeneralizationOpenAI's analog for scalable oversight (Burns et al., 2023): can a strong model, fine-tuned on labels from a weaker supervisor, generalise beyond the supervisor's capability? Experiments on NLP and chess tasks show it partially can; the residual quality gap motivates future work on supervising superhuman models.
Debate and Amplification (Scalable Oversight)Two proposed scalable-oversight protocols: debate pits two models against each other before a human judge; amplification recursively decomposes a difficult question into easier sub-questions that the supervisor can answer. Both aim to let weaker supervisors correctly oversee stronger AI by exploiting zero-sum pressure or composition.
Scalable Oversight and HonestyThe alignment sub-problem of producing reliable supervision for AIs that exceed their supervisors' capability. Proposals span protocols (debate, amplification, recursive reward modelling), training signals (constitutional AI, process-reward models), and the narrower but more tractable goal of honesty : aligning the model's outputs with its internal beliefs.
Model Organisms of MisalignmentAn Anthropic research programme (Hubinger et al., 2023): deliberately construct AI systems that exhibit specific, well-defined misalignment (e.g., deceptive reasoning, sandbagging, reward-hacking) to study the dynamics, detection, and removal of such behaviour — analogous to model organisms (mice, yeast) in biology.
Deceptive AlignmentA hypothesised failure mode (Hubinger et al., 2019) in which an AI deliberately behaves well during training and evaluation but pursues different goals at deployment. A deceptively aligned model is instrumentally aligned during oversight and misaligned otherwise, making detection exceptionally hard.
Sleeper AgentsHubinger et al. (2024) demonstrate that LLMs can be deliberately trained to have a backdoor — e.g., produce insecure code when a trigger string appears — and that standard safety training (RLHF, adversarial training, SFT on harmlessness) fails to remove the backdoor. Evidence that once deceptive alignment is present, it may persist through the standard post-training pipeline.
Feature Attribution (Integrated Gradients, SHAP)Post-hoc interpretability methods that attribute a model's prediction to its input features. Integrated Gradients (Sundararajan et al., 2017) integrates gradients along a path from a baseline to the input. SHAP (Lundberg & Lee, 2017) builds on Shapley values from cooperative game theory, giving a unique attribution with axiomatic fairness properties.
Monosemanticity and SAE FeaturesMonosemanticity is the idea that a single learned feature corresponds to a single interpretable concept rather than many unrelated ones. Sparse autoencoders are used to decompose dense neural activations into a wider sparse basis where such features are easier to identify.
Tool Use / Function Calling Benchmarks (BFCL, τ-bench)BFCL (Berkeley Function Calling Leaderboard, 2024) and τ-bench (2024) evaluate LLMs' ability to select, parameterise, and sequence API calls. BFCL is single-turn function-call accuracy; τ-bench is multi-turn dialogue in simulated customer-service environments with realistic state and policy constraints.
Agent Benchmarks (SWE-bench, GAIA, WebArena)SWE-bench tests whether a model can fix real GitHub issues in code repositories, GAIA tests general tool-using problem solving with automatically checked answers, and WebArena tests web-navigation agents in simulated sites. Together they measure software, reasoning, and browser-action competence rather than just one-shot text generation.
Matrix Rank and the Rank–Nullity TheoremThe rank of \( A \in \mathbb{R}^{m \times n} \) is the dimension of its column space, equal to the dimension of its row space. The rank–nullity theorem states \( \text{rank}(A) + \text{nullity}(A) = n \), linking how much a linear map preserves to how much it collapses — the foundation for invertibility conditions, low-rank adaptation, and OLS identifiability.
Matrix Norms: Frobenius, Spectral, NuclearThe three dominant matrix norms measure different notions of size: Frobenius \( \|A\|_F = \sqrt{\sum_{ij} A_{ij}^2} \) is the entrywise \( \ell_2 \); spectral (operator) \( \|A\|_2 = \sigma_{\max}(A) \) is the largest gain on unit vectors; nuclear \( \|A\|_* = \sum_i \sigma_i(A) \) is the convex surrogate for rank. Each appears in a distinct ML context: regularisation, Lipschitz bounds, and low-rank recovery respectively.
Positive (Semi-)Definite MatricesA symmetric matrix \( A \in \mathbb{R}^{n \times n} \) is positive semi-definite (PSD) if \( x^\top A x \ge 0 \) for all \( x \), and positive definite (PD) if strict for \( x \ne 0 \). Equivalent characterisations include non-negative eigenvalues and a Cholesky factorisation \( A = L L^\top \). PSD structure underlies covariance matrices, Gram matrices, convex Hessians, and every kernel method.
Moore–Penrose PseudoinverseFor any \( A \in \mathbb{R}^{m \times n} \) with SVD \( A = U\Sigma V^\top \), the Moore–Penrose pseudoinverse is \( A^+ = V\Sigma^+ U^\top \), where \( \Sigma^+ \) inverts the non-zero singular values and transposes. \( A^+ b \) is the minimum-norm least-squares solution to \( Ax = b \) — the canonical generalisation of \( A^{-1} \) for rectangular and rank-deficient matrices.
QR DecompositionAny \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \) factors as \( A = QR \), with \( Q \in \mathbb{R}^{m \times n} \) having orthonormal columns and \( R \in \mathbb{R}^{n \times n} \) upper triangular. It is the numerically preferred route to OLS, avoids squaring the condition number, and is the work-horse of least-squares solvers in every major numerical library.
Cholesky DecompositionEvery symmetric positive-definite matrix \( A \) factors uniquely as \( A = LL^\top \) with \( L \) lower triangular with positive diagonal. Cholesky is twice as fast as LU, numerically stable without pivoting, and the standard building block for Gaussian-process inference, Kalman filters, and sampling from multivariate normals.
Matrix Calculus and DifferentialsMatrix calculus expresses derivatives of scalar, vector, and matrix-valued functions of matrix inputs in compact form, avoiding entrywise index chasing. Working with the differential \( dL = \text{tr}(G^\top dX) \) identifies \( \nabla_X L = G \) directly and turns backpropagation into linear algebra.
Convex Optimization FundamentalsA convex optimization problem minimises a convex function over a convex feasible set. Its defining property: every local minimum is global. Convexity also gives polynomial-time algorithms with strong guarantees — the backdrop against which non-convex deep learning is understood as a deliberate departure.
KKT Conditions and Lagrangian DualityThe Karush–Kuhn–Tucker (KKT) conditions generalise \( \nabla f = 0 \) to constrained optimisation, giving necessary (and, for convex problems, sufficient) conditions for a minimum. Lagrangian duality transforms the constrained primal into a dual over multipliers; for convex problems with Slater's condition, the duality gap closes — the workhorse derivation behind SVMs, interior-point methods, and much of RL.
Newton's Method and Quasi-Newton (L-BFGS)Newton's method uses the second-order model \( f(x+\Delta) \approx f(x) + g^\top \Delta + \tfrac{1}{2}\Delta^\top H \Delta \) to take the step \( \Delta = -H^{-1} g \). It converges quadratically near a minimum but is impractical in high dimensions. Quasi-Newton methods (BFGS, L-BFGS) approximate \( H^{-1} \) from gradient histories, retaining super-linear convergence at the cost of a Hessian solve per step.
Proximal Gradient Methods (ISTA / FISTA)Proximal gradient methods solve objectives of the form \(f(x)+g(x)\) where \(f\) is smooth and \(g\) is nonsmooth but has an easy proximal operator. Each step does gradient descent on \(f\) and then a shrinkage- or projection-like update for \(g\), which is why the method underlies Lasso, sparse coding, and constrained convex optimization.
Conjugate Gradient MethodConjugate gradient (CG) solves \( Ax = b \) for symmetric PD \( A \) using only matrix-vector products, converging in at most \( n \) steps in exact arithmetic and much faster when the eigenvalue spectrum is clustered. CG underlies truncated-Newton optimisation, the natural-gradient in K-FAC, and large-scale Gaussian-process inference.
Random Variables, Expectation, and Variance (Axiomatic)A random variable \( X \) is a measurable map from a probability space \( (\Omega, \mathcal{F}, P) \) to \( \mathbb{R} \). Its expectation \( \mathbb{E}[X] = \int X\,dP \) and variance \( \text{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] \) are the first two moments. Beyond textbook formulas, this axiomatic view explains why expectations linearise sums, exist iff \( \mathbb{E}|X| < \infty \), and commute with limits under uniform integrability.
Bernoulli, Binomial, Categorical, and Multinomial DistributionsThe four atomic discrete distributions: Bernoulli \( (p) \) for a single binary trial, Binomial \( (n, p) \) for a sum of \( n \) such trials, Categorical \( (\pi) \) for a single \( K \)-class outcome (softmax target), and Multinomial \( (n, \pi) \) for counts across \( n \) such trials. They form the likelihood backbone of logistic regression, cross-entropy training, and count models.
Beta and Dirichlet DistributionsThe Beta \( (\alpha, \beta) \) is the conjugate prior for Bernoulli/Binomial likelihoods; the Dirichlet \( (\boldsymbol{\alpha}) \) is its \( K \)-class generalisation, conjugate to Categorical/Multinomial. Both live on probability simplices and make Bayesian updating a matter of adding counts to hyperparameters — the cleanest introduction to conjugate Bayesian inference and the scaffolding for LDA.
Poisson and Exponential DistributionsPoisson \( (\lambda) \) models counts of independent events in a fixed interval; Exponential \( (\lambda) \) models the waiting time between them. They are the two marginals of the Poisson process — one discrete, one continuous — and between them cover rare-event counts, waiting times, and the building blocks of survival analysis, rate modelling, and queueing theory.
Mutual Information and Conditional EntropyConditional entropy \( H(Y \mid X) \) measures the residual uncertainty in \( Y \) given \( X \). Mutual information \( I(X; Y) = H(Y) - H(Y \mid X) \) measures how much \( X \) reduces the uncertainty in \( Y \) — equivalently, the KL divergence from the joint \( p(X,Y) \) to the product of marginals \( p(X)p(Y) \). Together they quantify dependence, drive information-bottleneck theory, and define decision-tree splits.
Hypothesis Testing, p-values, and Statistical PowerA hypothesis test compares a null \( H_0 \) to an alternative \( H_1 \) by computing a test statistic and its tail probability under \( H_0 \) — the p-value. Statistical power is \( 1 - \beta \), the probability of rejecting \( H_0 \) when \( H_1 \) is true. For ML evaluation, these are the tools that separate "this model is better" from "this split was lucky".
Cramér–Rao Lower BoundThe Cramér–Rao bound states that any unbiased estimator \( \hat\theta \) of a parameter \( \theta \) has variance \( \text{Var}(\hat\theta) \ge 1/I(\theta) \), where \( I(\theta) \) is the Fisher information. It is the foundational efficiency bound of classical statistics and gives an immediate lower bound on the uncertainty of MLE-based procedures.
Gaussian Process RegressionA Gaussian process defines a distribution over functions such that any finite set of evaluations is jointly Gaussian. Given a kernel \( k \) and noisy observations, the posterior at a test point is \( \mathcal{N}(\mu_*, \sigma_*^2) \) with closed-form mean and variance. GP regression gives both predictions and calibrated uncertainty, costs \( O(n^3) \) for \( n \) training points, and is the Bayesian counterpart of kernel ridge regression.
VC Dimension and ShatteringThe Vapnik–Chervonenkis dimension of a hypothesis class \( \mathcal{H} \) is the largest number of points \( \mathcal{H} \) can shatter — label in every possible way. It controls PAC generalisation: if \( \text{VC}(\mathcal{H}) = d \), then with probability \( 1 - \delta \) all \( h \in \mathcal{H} \) satisfy \( L(h) \le \hat L(h) + O(\sqrt{d/n}) \). VC dimension explains why linear classifiers in \( \mathbb{R}^d \) need \( \Omega(d) \) examples and why simple hypothesis classes generalise.
Rademacher ComplexityThe empirical Rademacher complexity of a function class \( \mathcal{F} \) on data \( S = (z_1, \dots, z_n) \) is \( \hat{\mathfrak{R}}_S(\mathcal{F}) = \mathbb{E}_\sigma[\sup_{f \in \mathcal{F}} \tfrac{1}{n}\sum_i \sigma_i f(z_i)] \) — the expected ability of \( \mathcal{F} \) to correlate with random \( \pm 1 \) signs. It is the data-dependent workhorse of modern generalisation bounds, usually tighter than VC, and gives direct norm-based bounds for deep networks.
No-Free-Lunch TheoremThe No-Free-Lunch theorem says that averaged uniformly over all possible target functions, no learning algorithm outperforms any other. In machine learning it means performance gains must come from inductive bias that matches the structure of the problems we actually care about.
Goodhart's Law and Specification GamingGoodhart's Law says that once a measure becomes a target, optimizing it can break its usefulness as a proxy. In AI safety and ML systems this appears as specification gaming: the model finds ways to maximize the metric without achieving the intended goal.
t-SNE (t-Distributed Stochastic Neighbor Embedding)t-SNE embeds high-dimensional points into 2D/3D by matching a heavy-tailed joint \( Q \) in the embedding space to a Gaussian-kernel joint \( P \) over neighbourhoods in the input space, minimising \( D_{\text{KL}}(P\|Q) \). The heavy Student-t tail in \( Q \) solves the crowding problem; perplexity tunes neighbourhood size. t-SNE preserves local structure well but distorts global distances.
UMAP (Uniform Manifold Approximation and Projection)UMAP constructs a fuzzy topological representation of the data using local Riemannian metrics, then finds a low-dimensional embedding whose fuzzy topology matches. Its cross-entropy objective has distinct attractive and repulsive terms — easier to scale than t-SNE — and a theoretical motivation grounded in category theory and Riemannian geometry. In practice, UMAP preserves more global structure than t-SNE at similar or better speed.
DBSCAN (Density-Based Clustering)DBSCAN clusters points by density: a point is a core point if it has at least \( \text{minPts} \) neighbours within radius \( \varepsilon \); clusters are connected components of core-point neighbourhoods. Non-core points without core neighbours are labelled noise. Unlike \( k \)-means, DBSCAN does not need the number of clusters in advance and handles arbitrary cluster shapes, at the cost of \( \varepsilon \) tuning.
Linear and Quadratic Discriminant Analysis (LDA / QDA)LDA and QDA are generative classifiers: model class-conditionals \( p(x \mid y = k) = \mathcal{N}(\mu_k, \Sigma_k) \) and apply Bayes' rule. LDA assumes shared covariance \( \Sigma \) across classes, giving linear decision boundaries and shrinkage-like regularisation; QDA uses per-class \( \Sigma_k \), giving quadratic boundaries at the cost of \( K \times d^2 \) parameters. Both are the generative counterparts of logistic regression.
Matrix Factorisation and Collaborative FilteringMatrix factorisation decomposes a sparse user–item rating matrix \( R \approx UV^\top \) into low-rank user and item embeddings, learned by minimising squared error on observed entries with regularisation. This is the canonical collaborative-filtering recipe behind Netflix-style recommenders, the spiritual ancestor of modern embedding-based retrieval, and a clean worked example of low-rank learning.
Lottery Ticket HypothesisA dense randomly-initialised neural network contains subnetworks ("winning tickets") that — when trained in isolation with their original initialisation — match the full network's accuracy in the same number of steps. This Frankle–Carbin observation motivates one-shot and iterative magnitude pruning as search algorithms for sparse trainable subnetworks, reframing pruning as an initialisation search rather than a post-hoc compression.
Loss Landscape: Flat vs Sharp MinimaFlat minima (low curvature / small Hessian eigenvalues) generalise better than sharp minima (high curvature), empirically and via PAC-Bayes bounds. SGD's noise, large batch sizes, and the Sharpness-Aware Minimisation (SAM) optimiser all interact with this: small-batch SGD prefers flat minima, large-batch SGD falls into sharper ones, and SAM explicitly penalises sharpness during training.
Implicit Regularisation of SGDOver-parameterised networks trained by SGD generalise despite being able to fit pure noise — SGD's trajectory biases the solution toward specific minima. For linear models, gradient flow converges to the minimum-norm interpolator; for deep nets, SGD with small LR and moderate batch behaves like Bayesian inference with an implicit prior on flat minima. This implicit bias is why modern deep learning does not need explicit capacity control.
Multi-Armed Bandits: ε-Greedy, UCB, Thompson SamplingA multi-armed bandit is the simplest setting for exploration vs exploitation: \( K \) arms, each with unknown reward distribution; pull one per step; minimise cumulative regret. The three canonical strategies — ε-greedy, Upper Confidence Bound (UCB), and Thompson sampling — illustrate optimism, confidence-based exploration, and probability matching respectively, and generalise to contextual bandits and full RL.
Temporal-Difference Learning: TD(0), SARSA, TD(λ)Temporal-difference learning updates value estimates using the Bellman bootstrapped target \( R_t + \gamma V(S_{t+1}) \) rather than a full Monte Carlo return. TD(0) is the one-step instance; SARSA extends it on-policy to action-values; TD(λ) interpolates between TD(0) and Monte Carlo via eligibility traces. TD is the central learning rule behind Q-learning, DQN, and actor-critic.
Deep Q-Network (DQN)DQN parameterises \( Q(s, a; \theta) \) by a CNN and trains it by Q-learning on mini-batches sampled from a replay buffer, with a slowly-updated target network stabilising the bootstrapped target. Mnih et al. (2015) showed a single DQN architecture learning 49 Atari games from raw pixels at human-level on many — the empirical breakthrough that ignited modern deep RL.
Policy Gradient TheoremFor a stochastic policy \( \pi_\theta(a \mid s) \), the gradient of expected return \( J(\theta) = \mathbb{E}_{\tau \sim \pi_\theta}[G(\tau)] \) is \( \nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(A \mid S) \cdot Q^{\pi_\theta}(S, A)] \). No gradient flows through the environment dynamics — the theorem turns RL into a stochastic optimisation over policy parameters. It is the foundation of REINFORCE, actor-critic, PPO, TRPO, and GRPO.
Trust Region Policy Optimization (TRPO)TRPO performs policy updates by solving a constrained optimisation: maximise a surrogate advantage subject to \( \mathbb{E}[D_{\text{KL}}(\pi_\text{old} \| \pi_\theta)] \le \delta \). The KL trust region gives a monotonic-improvement guarantee and prevents collapse under function approximation. TRPO is solved by natural-gradient + line search; PPO is its first-order clipped approximation with near-identical performance at much lower cost.
BERT: Bidirectional Encoder PretrainingBERT (Devlin et al., 2019) is a Transformer encoder pretrained with masked language modelling (MLM) and next-sentence prediction (NSP), producing bidirectional contextual representations. Unlike GPT's causal, left-to-right pretraining, MLM sees both past and future tokens, making BERT the go-to encoder for classification, span extraction, and sentence embeddings. Base/Large variants (110M/340M params) dominated GLUE/SQuAD until decoder-only LLMs took over.
WordPiece, Unigram, and SentencePiece TokenisationWordPiece, Unigram, and SentencePiece are subword tokenization schemes used to balance vocabulary size against sequence length. WordPiece builds a vocabulary greedily, Unigram prunes a probabilistic vocabulary by likelihood, and SentencePiece is the language-agnostic toolkit that trains and applies these methods on raw text.
Prefix-Tuning and Prompt-TuningPrompt-tuning and prefix-tuning freeze the base model and learn only a small set of continuous prompt parameters. Prompt-tuning adds learned embeddings at the input, while prefix-tuning adds learned key-value prefixes inside each attention layer.
Adapter Layers (Houlsby-style)Houlsby-style adapters are small bottleneck MLPs inserted inside each Transformer block while the original backbone is frozen. They were the first clean PEFT recipe: add a few trainable parameters per layer, keep the base model unchanged, and specialize the model to new tasks cheaply.
Mixture of Depths (MoD)Mixture of Depths (Raposo et al., 2024) lets each token choose whether to go through the expensive self-attention + MLP stack at each layer, or to skip it via a residual. A small router predicts a saliency score; the top-\( k \) tokens per batch compute, the rest pass through. This per-token adaptive compute is the depth-axis counterpart of Mixture-of-Experts (width-axis) and substantially reduces FLOPs at matched quality.
GPU Memory Hierarchy and the Roofline ModelThe GPU memory hierarchy ranges from slow, large global memory to faster on-chip caches, shared memory, and registers. The roofline model explains performance by comparing arithmetic intensity with hardware limits: low-intensity kernels are memory-bound, while high-intensity kernels are compute-bound.
NCCL Collectives: All-Reduce, All-Gather, Reduce-ScatterNCCL (NVIDIA Collective Communications Library) implements ring and tree algorithms for GPU-to-GPU communication primitives — the building blocks of distributed deep learning. All-reduce sums gradients across workers; reduce-scatter + all-gather are the two halves of all-reduce, exploited by ZeRO and FSDP to shard parameters. Understanding these primitives is essential to reasoning about training scalability.
Arithmetic IntensityArithmetic intensity is the number of floating-point operations performed per byte of data moved from memory. In the roofline model it determines whether a kernel is memory-bound or compute-bound, which is why matmuls are efficient and elementwise ops often are not.
Triton GPU Kernel ProgrammingTriton (OpenAI, 2019) is a Python-embedded DSL for writing high-performance GPU kernels. It exposes tile-level primitives (block pointers, block-scoped arithmetic, automatic vectorisation) while hiding shared-memory management and thread-level scheduling. FlashAttention-2, Mamba kernels, many PyTorch 2 inductor-generated kernels, and most modern custom ops are written in Triton — fast enough to match hand-tuned CUTLASS with far less code.
LLaVA: Visual Instruction TuningLLaVA (Liu et al., 2023) connects a frozen CLIP vision encoder to a frozen LLM via a small learned projection, then instruction-tunes the combined model on GPT-4-generated multimodal dialogues. The recipe is minimal — a single linear (later two-layer MLP) projector — yet competitive with closed VLMs, establishing the canonical open VLM pattern: (pretrained vision encoder) + (learned bridge) + (pretrained LLM) + (visual instruction data).
Audio Language Models (AudioLM, MusicLM)Audio language models tokenise raw audio into discrete codes via neural audio codecs (SoundStream, Encodec), then model sequences of codes with a Transformer — the same next-token-prediction recipe as LLMs, applied to audio. AudioLM (Borsos et al., 2022) uses a two-level hierarchy of semantic and acoustic tokens for speech continuation; MusicLM extends to text-conditioned music generation.
HNSW (Hierarchical Navigable Small World)HNSW is the graph-based approximate nearest-neighbour (ANN) algorithm powering most production vector databases. It maintains a multi-layer proximity graph where each layer is a small-world graph at a different density; search descends from sparse top layers to dense bottom layers by greedy edge-following. Query cost is \( O(\log n) \); recall-latency trade-off is tunable per query.
ColBERT and Late-Interaction RetrievalColBERT (Khattab & Zaharia, 2020) represents each document as a bag of contextualised token vectors rather than a single vector, and scores against a query via MaxSim : for each query token, find its best-matching document token and sum the similarities. Late interaction preserves token-level granularity that pooling destroys, closing the quality gap between dense retrievers and cross-encoders at a fraction of the cost.
Graph Convolutional Network (GCN)Kipf & Welling's GCN (2017) applies a first-order spectral convolution on graphs: each node's representation is updated as a normalised sum of its neighbours' features, \( H^{(\ell+1)} = \sigma(\tilde D^{-1/2} \tilde A \tilde D^{-1/2} H^{(\ell)} W^{(\ell)}) \). The symmetric normalisation comes from a spectral argument; practically, it is the simplest and most widely-taught GNN layer, setting the template for all message-passing architectures.
Graph Attention Network (GAT)GAT (Veličković et al., 2018) replaces GCN's fixed degree-normalised aggregation with attention : each node learns per-edge weights via a shared attention mechanism. This gives inductive generalisation (no dependence on the full graph's degree matrix), handles heterogeneous neighbourhoods, and approaches Transformer-style flexibility — though at higher computational cost than GCN.
Active Inference and the Free-Energy PrincipleFriston's free-energy principle frames perception, learning, and action as minimising a single quantity — variational free energy, a KL-plus-accuracy bound on surprise. Active inference extends the principle to behaviour: agents select actions that minimise expected free energy, simultaneously seeking reward and information. It is a generative-model-based alternative to classical RL with tight links to variational inference, ELBO, and exploration–exploitation.
Wasserstein Distance & Optimal TransportThe \( p \)-Wasserstein distance \( W_p(\mu,\nu) = \inf_{\gamma \in \Pi(\mu,\nu)} \big( \mathbb{E}_{(x,y)\sim\gamma}\|x-y\|^p \big)^{1/p} \) measures the minimum cost of reshaping distribution \( \mu \) into \( \nu \). It underpins WGAN, flow matching, and a whole family of divergences that remain well-behaved when KL blows up.
f-Divergences (Unified View)For any convex \( f \) with \( f(1) = 0 \), the \( f \)-divergence \( D_f(P \| Q) = \mathbb{E}_Q[f(dP/dQ)] \) recovers KL (\( f = t \log t \)), reverse KL, Jensen–Shannon, total variation, \( \chi^2 \), Hellinger, and α-divergences as special cases. The variational (Fenchel) form underlies f-GAN and density-ratio estimation.
Itô Calculus & Stochastic Differential EquationsItô calculus extends ordinary calculus to processes driven by Brownian motion. An SDE \( dX_t = \mu(X_t, t)\,dt + \sigma(X_t, t)\,dW_t \) combines a drift and a diffusion term; Itô's lemma replaces the chain rule. This is the mathematical substrate of score-based diffusion models, flow matching, and neural SDEs.
Fokker–Planck & Probability-Flow ODEThe Fokker–Planck equation \( \partial_t p_t = -\nabla \cdot (f p_t) + \tfrac{1}{2} \nabla^2 : (g g^\top p_t) \) governs how the density of an SDE-driven process evolves. The probability-flow ODE shares these exact marginals with a deterministic vector field, enabling DDIM-style deterministic sampling and likelihood computation.
Reproducing Kernel Hilbert Spaces (RKHS) & the Representer TheoremAn RKHS is a Hilbert space of functions where evaluation at a point can be written as an inner product with a kernel function. The representer theorem says that many regularized empirical-risk problems in an RKHS have solutions that are finite sums of kernel evaluations at the training points, making kernel methods practical.
Mercer's Theorem & Kernel Feature MapsMercer's theorem shows that a continuous positive-semidefinite kernel can be expanded in nonnegative eigenvalues and orthogonal eigenfunctions. This makes kernel functions equivalent to inner products in a possibly infinite-dimensional feature space and motivates the kernel trick.
Natural Gradient & Fisher–Rao GeometryThe natural gradient \( \tilde\nabla_\theta \mathcal{L} = F(\theta)^{-1} \nabla_\theta \mathcal{L} \) preconditions the Euclidean gradient by the inverse Fisher information matrix, yielding steepest descent under the KL-divergence metric on the statistical manifold. It underlies K-FAC, Shampoo, TRPO's trust region, and the original motivation for reparameterisation-invariant optimisation.
Hamiltonian Monte Carlo (HMC) & NUTSHMC augments the target distribution with an auxiliary momentum variable and simulates Hamiltonian dynamics so that proposals move long distances while staying on near-constant-energy shells. It explores complex posteriors dramatically faster than random-walk Metropolis; the No-U-Turn Sampler (NUTS) adapts the integration length automatically.
Metropolis–Hastings AlgorithmThe canonical MCMC recipe: propose \( \theta' \sim q(\theta' \mid \theta) \), accept with probability \( \min(1, \pi(\theta')q(\theta\mid\theta')/[\pi(\theta)q(\theta'\mid\theta)]) \). Produces a Markov chain with stationary distribution \( \pi \) for any valid proposal, turning intractable posterior sampling into a correctness-guaranteed iterative procedure.
Gibbs SamplingA special case of Metropolis–Hastings: cycle through variables, sampling each from its full conditional \( p(\theta_j \mid \theta_{-j}, D) \). When the conditionals are tractable (conjugate priors, mixture models, LDA), Gibbs is simple, always accepts, and has been the workhorse of Bayesian inference for four decades.
Concentration Inequalities (Hoeffding, Bernstein, McDiarmid)High-probability bounds on how far a sum or function of independent random variables can deviate from its mean. Hoeffding uses boundedness, Bernstein exploits known variance for a tighter bound, and McDiarmid handles functions whose value changes little when any single argument changes — the workhorses behind PAC / generalization proofs.
PAC-Bayes Generalization BoundsPAC-Bayes bounds the generalization gap of a stochastic classifier \( Q \) by a Kullback–Leibler term against a data-independent prior \( P \): with probability \( 1 - \delta \), \( \mathbb{E}_{h\sim Q}[R(h)] \le \mathbb{E}_{h\sim Q}[\hat R(h)] + O(\sqrt{\text{KL}(Q\|P)/n}) \). Used for non-vacuous bounds on overparameterised networks.
XGBoost, LightGBM, and CatBoostThree production gradient-boosted-decision-tree implementations with distinct tree-construction strategies: XGBoost does level-wise exact/approximate splits with second-order Taylor objective; LightGBM uses histogram-based leaf-wise growth and GOSS subsampling; CatBoost uses ordered boosting to avoid target leakage with categorical features.
Hierarchical Clustering & LinkageBuilds a tree (dendrogram) of clusters either bottom-up (agglomerative: start with singletons, merge closest pairs) or top-down (divisive). The linkage criterion — single, complete, average, Ward — defines distance between clusters and dictates the cluster shapes the algorithm prefers.
Spectral Clustering & the Graph LaplacianConstruct a similarity graph over data points, embed each point into \( \mathbb{R}^k \) using the \( k \) smallest eigenvectors of the graph Laplacian, and run k-means in that space. The Laplacian's spectrum encodes cluster structure as low-frequency eigenmodes — works for non-convex clusters where Euclidean k-means fails.
Non-Negative Matrix Factorization (NMF)Factorise a nonnegative matrix \( V \approx W H \) with \( W, H \ge 0 \) entrywise. The nonnegativity constraint yields parts-based, interpretable components (topic–word, basis–image) and distinguishes NMF from PCA, whose sign-free components are typically holistic and hard to name.
Independent Component Analysis (ICA)ICA separates a linear mixture \( x = A s \) into statistically independent non-Gaussian sources by finding a de-mixing matrix \( W \) that maximises the non-Gaussianity of \( y = W x \). Classical application: the cocktail-party problem. Key distinction from PCA: maximises independence, not variance.
Kernel PCAPrincipal component analysis in the implicit feature space of a positive-definite kernel \( k(x, y) \). Eigendecomposes the centred Gram matrix \( K \) rather than the data covariance; recovers non-linear principal directions without ever instantiating the feature map. Used for non-linear dimensionality reduction and feature extraction.
Isomap & Locally Linear Embedding (LLE)Two classical manifold-learning algorithms: Isomap replaces Euclidean distances with geodesic distances on a \( k \)-NN graph and applies MDS; LLE reconstructs each point from its neighbours' linear weights and finds a low-dimensional embedding that preserves those weights. Both set the conceptual stage for t-SNE and UMAP.
Conditional Random Fields (CRF)A discriminative model of \( p(y \mid x) = \tfrac{1}{Z(x)} \exp \sum_k \lambda_k f_k(y, x) \) over structured outputs. Linear-chain CRFs add sequence-level constraints on top of per-token scores, enabling tractable training via forward–backward and Viterbi decoding — still the backbone of NER/tagging heads above neural encoders.
Bayesian Networks & Directed Graphical ModelsA Bayesian network is a DAG \( G \) over variables whose joint factorises as \( p(x) = \prod_i p(x_i \mid \text{pa}_G(x_i)) \). D-separation reads conditional independences off the graph; parameter learning uses MLE under complete data, EM under latent variables. The mathematical foundation of structured probabilistic modelling.
Belief Propagation (Sum–Product Algorithm)A message-passing algorithm that computes exact marginals on tree-structured graphical models in linear time and approximate marginals (loopy BP) on general graphs. Each node sends summarising messages to neighbours; final beliefs equal the product of incoming messages.
Variational Autoencoder (VAE)A latent-variable generative model trained by maximising the ELBO \( \mathcal{L}(x) = \mathbb{E}_{q_\phi(z\mid x)}[\log p_\theta(x\mid z)] - D_{\text{KL}}(q_\phi(z\mid x)\,\|\,p(z)) \). The reparameterisation trick makes the encoder \( q_\phi \) differentiable; the decoder \( p_\theta \) learns to reconstruct \( x \) from latent codes \( z \sim \mathcal{N}(0, I) \).
β-VAE & Disentanglementβ-VAE replaces the ELBO's KL term with a weighted \( \beta \cdot D_{\text{KL}} \). Values \( \beta > 1 \) push the encoder toward an isotropic prior, encouraging each latent dimension to capture one independent factor of variation — the original disentanglement recipe.
Normalizing Flows (RealNVP, Glow)Invertible neural networks \( f_\theta: \mathbb{R}^d \to \mathbb{R}^d \) with tractable Jacobian determinant. The change-of-variables formula \( \log p_X(x) = \log p_Z(f(x)) + \log |\det J_f(x)| \) gives exact likelihood; sampling runs \( f^{-1} \). RealNVP and Glow use coupling layers to make both directions \( O(d) \) per step.
Autoregressive Flows (MAF & IAF)Flows in which the \( i \)-th output depends only on previous inputs \( x_{<i} \), giving a triangular Jacobian. MAF (masked autoregressive flow) has fast density evaluation but slow sampling; IAF (inverse autoregressive flow) is the mirror image — fast sampling, slow density. Both are cornerstones of modern density estimation.
Energy-Based Models (EBM)A generative model \( p_\theta(x) = \exp(-E_\theta(x))/Z(\theta) \) defined by a scalar energy \( E_\theta \). The intractable normaliser \( Z(\theta) = \int e^{-E_\theta(x)} dx \) precludes direct MLE; training uses contrastive divergence, score matching, or noise-contrastive estimation to approximate it.
Restricted Boltzmann Machines (RBM)A bipartite EBM over visible and hidden binary units with energy \( E(v, h) = -v^\top W h - b^\top v - c^\top h \). Conditional independence within each layer gives closed-form conditionals \( p(h\mid v), p(v\mid h) \); Hinton's Contrastive Divergence trains them and the RBM stack forms a deep belief net.
Noise-Contrastive Estimation (NCE)Learn an unnormalised model \( \tilde p_\theta(x) \) by training a binary classifier to distinguish data samples from noise samples. The classifier's logit becomes \( \log \tilde p_\theta(x) - \log q_{\text{noise}}(x) \), so the partition function is absorbed into a learnable constant. Foundation of word2vec's negative sampling and of InfoNCE contrastive learning.
Score-Based SDEs (Continuous-Time Diffusion)Song et al. (2021) showed that discrete-time DDPM and noise-conditional score models are both limits of a continuous-time SDE \( dx = f(x,t)dt + g(t)dW \). The unified framework gives a reverse-time SDE and a probability-flow ODE that share marginals, enabling flexible samplers (Euler, Heun, DPM-Solver) and exact likelihoods.
GAN Family: WGAN, StyleGAN, BigGANThree architectural and objective milestones: WGAN uses the Kantorovich–Rubinstein dual of \( W_1 \) as a smoother critic, StyleGAN introduces AdaIN-controlled style injection for image generation, BigGAN scales class-conditional GANs to 512×512 with orthogonal regularisation and truncation tricks.
U-Net ArchitectureA fully-convolutional encoder–decoder with symmetric skip connections between contracting and expanding paths. Designed for biomedical segmentation; now the standard backbone of Stable Diffusion and most pixel-to-pixel models because skip connections preserve spatial detail across downsampling.
Semantic & Instance SegmentationSemantic segmentation assigns a class label to every pixel; instance segmentation further distinguishes object instances (two cats become two masks). Panoptic segmentation unifies them: one label per pixel with 'thing' vs 'stuff' classes. Backbones: FCN, DeepLab, Mask R-CNN, DETR/Mask2Former.
Object Detection: R-CNN → Faster R-CNN → DETRR-CNN ran a CNN classifier on externally-proposed regions; Fast R-CNN shared backbone features across proposals; Faster R-CNN introduced a learned Region Proposal Network; DETR replaced the entire region-proposal pipeline with a transformer that predicts a fixed set of boxes via bipartite matching.
YOLO Family (v1 – v10)Single-stage detectors that divide the image into a grid and predict bounding boxes and class probabilities directly from a single CNN forward pass. YOLOv1 was real-time but coarse; YOLOv3–v10 progressively adopted anchor boxes, FPN, CSP blocks, decoupled heads, and finally anchor-free / NMS-free designs for edge deployment.
Neural Radiance Fields (NeRF) & 3D Gaussian SplattingNeRF encodes a 3-D scene as a continuous function \( (x, y, z, \theta, \phi) \to (\text{colour}, \text{density}) \) queried along camera rays and volume-rendered into pixels. 3D Gaussian Splatting replaces the implicit MLP with an explicit set of anisotropic Gaussians rasterised in real time.
PointNet & 3D Deep Learning on Point CloudsPointNet processes an unordered point set by applying a shared MLP to each point, then pooling across points with a symmetric function (max-pool). Permutation-invariant by construction; PointNet++ adds local-region hierarchies to capture geometric structure.
Neural Ordinary Differential EquationsA neural ODE defines the hidden-state evolution as \( dh/dt = f_\theta(h, t) \), integrated by a black-box ODE solver. Training uses the adjoint method to back-propagate at constant memory regardless of solver depth. Connects residual networks to continuous flows and underlies continuous normalising flows and flow matching.
Pointer NetworksA seq2seq architecture whose decoder outputs indices into the input sequence via attention weights (a 'pointer') rather than tokens from a fixed vocabulary. Ideal for combinatorial problems whose output vocabulary depends on the input (convex hull, TSP, sorting) and for extractive QA / span prediction.
Siamese Networks & Metric LearningTrain a shared encoder so that semantically similar inputs map to nearby embeddings (contrastive / triplet loss) or that query-key scores reflect similarity directly. Used in face verification, signature matching, image retrieval; the conceptual parent of SimCLR, CLIP, and BiEncoder retrieval.
Highway NetworksPredecessor to ResNet: a gated skip connection \( y = H(x) \cdot T(x) + x \cdot (1 - T(x)) \), where \( T(x) \in [0,1] \) is a learned transform gate. Enabled training of 100+ layer networks before residual connections simplified the construction in ResNet.
Capsule NetworksHinton's alternative to CNN pooling: neurons are grouped into 'capsules' whose vector output encodes both existence and pose of an entity. Dynamic routing by agreement replaces max-pool, so each capsule decides which higher-level capsule to vote for based on agreement. Historically significant; practically superseded by transformers.
Memory-Augmented TransformersTransformer variants that extend effective context length with an external memory — Recurrent Memory Transformers (RMT) pass summary tokens across chunks, Memorizing Transformers retrieve past kNN keys, Infini-attention compresses the tail of context into a linear-attention state. A bridge between fixed-context Transformers and sequence models with unbounded memory.
Set Transformer & Deep Sets (Permutation Invariance)Deep Sets: any permutation-invariant function on sets equals \( \rho(\sum_i \phi(x_i)) \) for learnable \( \phi, \rho \). Set Transformer replaces the sum with self-attention via Induced Set Attention Blocks, giving element-wise interactions while remaining permutation-equivariant.
Sharpness-Aware Minimization (SAM)Minimise a loss whose value is worst-case over a small \( \rho \)-ball of weight perturbations: \( \min_\theta \max_{\|\varepsilon\| \le \rho} \mathcal{L}(\theta + \varepsilon) \). The ascent step \( \varepsilon^\star \approx \rho \, \nabla \mathcal{L}/\|\nabla \mathcal{L}\| \) biases training toward flat minima, improving generalisation across ViT, ResNet, and LLM finetuning.
Lion OptimizerA momentum-only optimizer discovered by neural search: updates are \( \theta \leftarrow \theta - \eta \, \text{sign}(\beta_1 m + (1-\beta_1) g) \). Uses only the sign of a momentum estimate — no second moment, half the state of AdamW, often matches or beats it on large-scale training with smaller learning rates.
Shampoo & K-FAC PreconditionersShampoo and K-FAC are second-order-inspired optimizers that precondition gradients with matrix or blockwise curvature information instead of only per-parameter learning rates. They aim to converge in fewer steps than Adam or SGD, especially in large-batch training where curvature estimates are more stable.
AdaFactor OptimizerA memory-efficient Adam variant. Replaces the full second-moment matrix with row and column running averages (a rank-1 factorisation): memory is \( O(m + n) \) instead of \( O(mn) \). Used by T5, Meta's BlenderBot, and many large-scale models to free memory for bigger batches and longer contexts.
Meta-Learning: MAML & ReptileMAML learns an initialisation \( \theta \) such that one or a few SGD steps on a new task yield good performance — formally \( \min_\theta \sum_\tau \mathcal{L}_\tau(\theta - \alpha \nabla \mathcal{L}_\tau(\theta)) \). Reptile is a first-order simplification that moves \( \theta \) toward per-task adapted parameters. Influential early 'learn to learn' recipes, since absorbed into prompt-based few-shot learning.
Focal Loss & Class-Imbalance ObjectivesFocal loss \( \text{FL}(p_t) = -(1-p_t)^\gamma \log p_t \) down-weights the loss contribution of confidently-classified easy examples, focusing gradient on hard ones. Designed for extreme foreground-background imbalance in one-stage detection; widely used in segmentation and long-tailed classification.
InfoNCE & NT-Xent Contrastive LossesInfoNCE maximises a mutual-information lower bound by classifying a positive pair against \( k \) negatives: \( \mathcal{L} = -\log \exp(s^+) / \sum_i \exp(s_i) \). NT-Xent is InfoNCE with temperature-scaled cosine similarities. Drives SimCLR, MoCo, CLIP, and most modern self-supervised representation learning.
Gradient Accumulation & Micro-BatchingSplit a large effective batch into \( k \) micro-batches; accumulate their gradients in a buffer, then step once. Decouples statistical batch size from hardware batch size, enabling \( N\cdot k \) effective batch without a \( k\times \) memory blow-up. Essential for training large models on any GPU.
Expected Calibration Error (ECE) & Reliability DiagramsMeasures miscalibration: \( \text{ECE} = \sum_b (|B_b|/n) \, |\text{acc}(B_b) - \text{conf}(B_b)| \). Predictions are bucketed by confidence; the gap between mean confidence and accuracy in each bucket sums to ECE. Deep nets are typically over-confident; temperature scaling restores calibration.
Platt Scaling & Isotonic RegressionPlatt scaling calibrates a classifier by fitting a one-dimensional logistic regression from raw scores to probabilities on a validation set. Isotonic regression is a more flexible monotonic alternative that can fit non-sigmoid calibration curves, but it usually needs more calibration data to avoid overfitting.
Bayesian Deep Learning: MC Dropout & Deep EnsemblesMC dropout estimates predictive uncertainty by keeping dropout active at test time and averaging many stochastic forward passes. Deep ensembles train several independently initialized models and usually give stronger uncertainty estimates, at higher training and serving cost.
Conformal PredictionA distribution-free procedure for turning any point predictor into a prediction set with guaranteed finite-sample coverage \( 1 - \alpha \) under exchangeability. Requires only a scoring function and a calibration set; no assumption on the underlying model or data distribution.
Paired Bootstrap & McNemar's Test for Model ComparisonTwo non-parametric procedures for deciding whether model A beats model B with statistical significance. Paired bootstrap resamples matched predictions to estimate a confidence interval on the metric difference. McNemar's test uses a chi-squared on the \( 2\times 2 \) contingency table of agreement / disagreement.
Data Curation & Quality Filters (FineWeb, Dolma)Modern pretraining pipelines filter terabytes of web data through language ID, heuristic rules (repetition, punctuation ratios), classifier-based quality scoring, and toxicity / PII removal. The FineWeb and Dolma recipes document which filters mattered — often delivering per-token quality gains equivalent to 2–3× scale-up.
MinHash & LSH for Large-Scale DeduplicationMinHash approximates Jaccard similarity between sets, and locality-sensitive hashing uses that approximation to quickly find near-duplicates. In ML data pipelines this is used to deduplicate documents or examples at very large scale.
Synthetic Data Generation for Post-TrainingModern instruction-tuning and RL-based alignment rely on LLM-generated synthetic data: self-instruct / Evol-Instruct expand seed prompts, teacher models produce high-quality completions, and process-reward models validate chain-of-thought steps. The backbone of the post-ChatGPT post-training stack.
Continual Pretraining & Mid-TrainingContinue pretraining an existing base model on a domain or task-focused corpus (code, math, a new language) before final post-training. Achieves domain gains that would cost 10× more to obtain by fine-tuning alone. Sits between pretraining and SFT in modern recipes.
Langevin Dynamics & MALALangevin MCMC treats gradient noise as deliberate: proposals drift along \( -\nabla \log \pi(\theta) \) plus Gaussian perturbation so the chain targets \( \pi \). Metropolis-adjusted Langevin (MALA) corrects discretisation bias with an MH acceptance; unadjusted Langevin (ULA) trades bias for simplicity and scales to big data via stochastic gradients (SGLD).
PixelCNN / PixelCNN++Autoregressive image models that factor \( p(x) = \prod_i p(x_i \mid x_{1:i-1}) \) with masked convolutions so each pixel sees only pixels above and to the left. Tractable likelihood and sharp samples; PixelCNN++ improves expressive conditioners (e.g. gated activations, horizontal/vertical stacks).
Self-Rewarding Language ModelsTrain a model to both generate responses and judge them, using the same weights. Each iteration: generate a pool of candidates, self-rate them, extract preference pairs, DPO-train on the pairs. Recursive improvement without external reward data; bottlenecks surface around judgement quality and diversity collapse.
Long-Context Data Recipes (RULER, Needle Variants)Extending effective context beyond 128k requires (a) RoPE-scaling or position-interpolation to keep positional encodings sane, (b) a continued-pretraining dataset with real long documents and synthetic stitched tasks, and (c) evaluation beyond simple needle-in-a-haystack — RULER adds multi-needle, multi-hop, and aggregation subtasks that expose superficial-match shortcuts.
Tokenizer Training Dynamics & Vocab SizingTokenizer design trades vocabulary size against sequence length and changes both compute cost and what patterns the model can represent cleanly. BPE, unigram, and byte-level schemes make different compromises, especially for code, multilingual text, and rare domain terms.
ORPO (Odds-Ratio Preference Optimization)A reference-model-free preference-optimisation objective that combines SFT and preference learning in one loss: \( \mathcal{L}_{\text{ORPO}} = \mathcal{L}_{\text{SFT}} - \lambda \log \sigma(\log \text{odds}(y_w) - \log \text{odds}(y_l)) \). Eliminates DPO's reference-model requirement, halving training memory.
SimPO (Simple Preference Optimization)Reference-free preference objective that replaces DPO's reference ratio with a length-normalised policy log-likelihood: \( r(x, y) = \log \pi_\theta(y\mid x) / |y| \). Adds a margin \( \gamma \) to the preferred response. Simpler than DPO, matches or beats it, and length-normalisation reduces verbosity exploitation.
IPO (Identity Preference Optimization)Replaces DPO's sigmoid objective with a squared-error criterion on preference probabilities: \( \mathcal{L}_{\text{IPO}} = \mathbb{E}[(h_\theta(y_w, y_l) - 1/(2\beta))^2] \). Prevents DPO's tendency to over-separate preferred and rejected responses on easy pairs, reducing overfitting and improving generalisation.
RLVR: RL from Verifiable RewardsTrain a reasoning policy with pure RL signals from tasks whose answers are automatically verifiable — math (exact match), code (unit-test execution), proof (checker), chess (engine eval). No preference model needed. The method behind DeepSeek-R1-Zero and the o1-style long-CoT reasoning families.
Reward Hacking & Specification Gaming in RLHFWhen a learned reward model is a proxy for human preference, RL optimisation finds adversarial inputs that maximise reward without matching the true objective — verbose apologies, sycophancy, confident wrong answers, format exploitation. Goodhart's law in practice. Mitigations range from KL penalty and reward normalisation to process supervision and debate.
Refusal Training & Harmlessness ObjectivesTeach a model to decline unsafe or out-of-policy queries while remaining helpful on benign ones. Typical recipe: SFT on curated refusal examples, preference optimisation with harm-labelled pairs, red-team-driven iteration. Badly done, this causes over-refusal (the 'brittle helpfulness' regime); done well, it achieves high refusal rate with minimal utility loss.
Graph of ThoughtsGeneralises chain-of-thought and tree-of-thought to an arbitrary DAG of reasoning nodes, each produced by a prompt. Enables aggregation (merge parallel solutions), refinement loops, and reuse of intermediate results. Useful for problems where branch-and-combine dominates (sorting, constraint satisfaction, multi-step planning).
Toolformer & Learned Tool UseToolformer trains a language model to decide when to call external tools and what arguments to send without requiring human demonstrations of tool use. It keeps tool calls only when the returned result improves the continuation, making tool use a self-supervised learning signal.
Plan-and-Solve PromptingTwo-stage prompt pattern: first ask the model to produce a plan (step-by-step outline), then execute the plan step by step. Improves over naive zero-shot CoT on arithmetic and multi-hop reasoning by forcing explicit decomposition. Contrasts with ReAct's interleaved thought-action loop.
Constrained Decoding: Grammars, JSON Mode, RegexGrammar-constrained decoding masks any next token that would violate a target grammar or schema. This guarantees outputs such as valid JSON, XML, or regex-matching strings by restricting generation to the language accepted by a finite-state machine or pushdown automaton.
Model Context Protocol (MCP)Model Context Protocol is an open client-server protocol for connecting models and agents to external tools, resources, and prompts through a common interface. It standardizes capability discovery and tool invocation so one MCP-aware client can talk to many different servers without bespoke integrations.
Agentic Workflows & Multi-Agent OrchestrationSystems that compose multiple LLM calls — planner, executor, critic, tool-user — into an end-to-end workflow. Patterns range from ReAct loops to fixed DAGs (LangGraph) to role-playing ensembles (AutoGPT, BabyAGI). Success requires careful handoff design, termination criteria, and cost control.
FlashAttention-2 and FlashAttention-3FlashAttention-2 and FlashAttention-3 are follow-on attention kernels that keep exact attention outputs while running much faster through better tiling, parallelism, and data movement. FA-2 improves work partitioning on modern GPUs, while FA-3 adds Hopper-specific asynchronous pipelines and low-precision support.
Medusa & EAGLE Speculative-Decoding HeadsSpeculative decoding with learned draft heads instead of a separate draft model. Medusa adds \( K \) small lightweight heads on the base model predicting future tokens; the base verifies tree-hypotheses in one forward pass. EAGLE models the residual stream directly and achieves 3–4× speedup with a tiny draft network.
Lookahead DecodingExact speculative-like decoding without any draft model: maintain a running n-gram 'Jacobi window' that proposes multiple tokens ahead in parallel; verify in one pass. Lossless — outputs match greedy decoding exactly — and requires no extra training. Trades increased per-step compute for fewer sequential steps.
Radix / Prefix-Cache Attention (SGLang)Share the KV cache across requests that start with a common prompt prefix. Store prefix trees keyed by token sequence; on a new request, find the longest matching prefix in the cache and reuse it. Cuts prefill latency and memory use for chat applications with shared system prompts or few-shot contexts.
Quantized KV Cache (int4 / int8 / KIVI)Store the KV cache at lower precision — int8 or int4 — instead of fp16. Halves or quarters the memory footprint of long contexts at negligible quality cost. Different quantisation per key / value (K usually int8, V int4 via grouping) and per-head asymmetric scales are the main tricks.
Continuous vs Static BatchingStatic batching groups requests before a forward pass and runs them to completion together — tail latency is set by the slowest request. Continuous batching (Orca, vLLM) evicts finished requests mid-step and admits new ones each iteration, keeping GPU utilisation high and tail latency bounded. Default in production LLM serving.
Transcoders & Sparse CrosscodersTranscoders and sparse crosscoders are interpretability models that learn sparse dictionaries linking features across layers rather than explaining one layer in isolation. They are used to trace how a concept is transformed, preserved, or split as it moves through a network.
Causal Scrubbing & Mediation AnalysisRigorous protocols for validating interpretability hypotheses. Causal scrubbing replaces the hypothesised-irrelevant computations with samples from a distribution that should preserve the output; mediation analysis tests whether a candidate component mediates the causal effect of an input on an output. Tools for turning 'this feature looks meaningful' into falsifiable claims.
Circuit Discovery Pipelines (ACDC, Attribution Patching)Automated methods to locate the minimal sub-graph of attention heads and MLP components responsible for a given behaviour. ACDC greedily ablates edges in a causal graph, keeping only those whose removal degrades the behaviour; attribution patching approximates this with a single forward-backward pass per hypothesis.
Representation Engineering & the Refusal DirectionShift model behaviour by directly manipulating residual-stream activations along interpretable directions. The 'refusal direction' (Arditi et al. 2024) is a single direction in activation space whose ablation jailbreaks open-weight chat models, and whose injection forces refusal — evidence that safety training installs a shallow, targeted feature.
Concept Erasure & Null-Space ProjectionRemove a protected concept (gender, ethnicity, refusal, a specific memory) from representations by iteratively projecting activations onto the null space of linear classifiers for that concept. Achieves provable linear guarding of downstream use against the erased attribute, with bounded utility loss.
Differential Privacy & DP-SGDFormal guarantee that an algorithm's output distribution barely changes if any single training example is replaced. DP-SGD achieves \( (\varepsilon, \delta) \)-DP by clipping per-example gradients and adding calibrated Gaussian noise. Central to privacy-preserving ML training on sensitive data.
Federated Learning (FedAvg)Train a shared model across many clients (phones, hospitals) without centralising data. Each round: clients train locally for a few epochs, server averages their weight updates. Introduces non-IID-data, communication-cost, and privacy / security challenges absent in centralised training.
Membership Inference Attacks (MIA)Determine whether a specific example was in the training set of a deployed model. Attacks exploit loss / confidence gaps between seen and unseen examples — trained models are typically more confident on memorised training points. A baseline for privacy leakage in ML systems.
Model Stealing & Extraction AttacksModel stealing attacks recover a useful copy of a deployed model by querying it and training a substitute on the outputs. Extraction attacks go further and try to recover hidden parameters, decision rules, or embeddings directly, which matters for both proprietary models and privacy-sensitive systems.
Data Poisoning & Backdoor AttacksInsert malicious training examples so the model learns a targeted behaviour — misclassification on a trigger pattern, backdoored refusal bypasses, or degraded accuracy on specific classes. BadNets demonstrated pixel-trigger backdoors; modern LLM poisoning targets alignment-layer susceptibilities and pretraining data.
LLM Watermarking (Kirchenbauer et al.)Embed a statistical signature into generated text that is invisible to humans but detectable by an algorithm with the watermarking secret. Kirchenbauer et al. (2023) partition the vocabulary into a pseudo-random green / red list per step, biasing generation toward green; later detection uses a \( z \)-test on green-token frequency.
Dangerous-Capability Evaluations (Bio, Cyber, Persuasion, Autonomy)Dangerous-capability evaluations are targeted tests for whether a model can meaningfully assist with high-consequence harms such as bio misuse, cyber offense, persuasive manipulation, or autonomous scheming. They are used as deployment-gating evidence because ordinary benchmark gains do not tell you whether a model has crossed a safety-relevant threshold.
Prompt Injection: Taxonomy & DefencesAdversarial instructions embedded in model-accessible content — tool outputs, retrieved documents, emails — that override the user's original task. Direct (in user prompt) vs indirect (in external content). Defences include input filtering, dual-model separation, and structured prompt templates; none is a complete solution.
CTC Loss & RNN-TransducerTwo objectives for training sequence-to-sequence models when alignment between input and output frames is unknown. CTC sums over all alignment paths with blank symbols; RNN-T decomposes into a prediction network and joint network to model output-length independently. Backbones of modern ASR pipelines.
Conformer ArchitectureA hybrid CNN-plus-attention block for speech recognition: each Conformer layer combines multi-head self-attention (global), depthwise convolutions (local), and sandwiched feed-forward modules. Outperforms pure Transformer and pure CNN on LibriSpeech and became the de facto encoder for production ASR.
Text-to-Image: DALL-E Lineage & ImagenAutoregressive (DALL-E 1, Parti) vs diffusion (DALL-E 2, DALL-E 3, Imagen, Stable Diffusion, Flux) lineages for prompt-to-pixel generation. DALL-E 3 uses a specialised caption-rewriting stage; Imagen emphasises text-encoder scale (T5-XXL) as the dominant quality lever.
Unified Multimodal Models (GPT-4o / Gemini any-to-any)Single models that process and generate multiple modalities — text, image, audio, video — through a shared backbone with per-modality tokenisers. Native multimodal training yields far richer cross-modal reasoning than cascaded pipelines: image understanding in context of speech, audio generation from visual cues, unified embeddings.
Video Diffusion (Sora, Veo, Gen-3)Extend image-diffusion recipes to video with 3D patch embeddings, temporal attention, and long-context handling. Sora (OpenAI), Veo (Google), and Gen-3 (Runway) train DiT-style transformers over space-time patches of 1–60 second clips, conditioning on rich text captions for controllable generation.
Mean Field Theory of Neural NetworksMean-field theory studies very wide neural networks by tracking distributions of parameters or activations instead of individual weights. It yields clean scaling limits for training dynamics and feature learning, and helps distinguish true feature-learning regimes from the lazy-training NTK regime.
Information Bottleneck TheoryInformation Bottleneck theory studies representations that preserve information about the target while compressing information about the input, often through a trade-off like \(I(Z;Y) - eta I(Z;X)\). It is a useful lens on representation learning and generalization, though its direct explanatory power for deep networks remains debated.
Stability and GeneralizationAn algorithm is uniformly \( \beta \)-stable if replacing one training point changes its output's loss by at most \( \beta \). Bousquet & Elisseeff (2002) proved that \( \beta \)-stability bounds the generalization gap by \( O(\beta + 1/\sqrt{n}) \); Hardt, Recht & Singer (2016) showed SGD on smooth losses is \( O(T/n) \)-stable, giving the first algorithm-dependent generalization bound for deep learning that grows with training time.
Algorithmic Alignment TheoryA neural architecture generalises better on a reasoning task when its computational structure aligns with the algorithm that solves the task. Xu et al. (2020) formalise sample complexity in terms of the number of network modules that must be learned and the per-module learnability, predicting that GNNs (multi-step message passing) align with dynamic-programming algorithms while plain MLPs do not.
Spectral Bias of Neural NetworksSpectral bias is the tendency of gradient-trained neural networks to learn low-frequency or smooth components of a target function before high-frequency ones. This helps explain why neural nets often fit coarse structure early and fine detail later.
Neural CollapseAt the terminal phase of training (TPT) — long after zero training error — the last-layer features and classifier weights of a deep classifier converge to a highly symmetric configuration: per-class feature means form a Simplex Equiangular Tight Frame (ETF), within-class variability collapses to zero, classifier weights align with the class means, and prediction reduces to nearest-class-centre. Papyan, Han & Donoho (2020) established this as a robust empirical phenomenon across architectures and datasets.
Mode Connectivity in Loss LandscapesMode connectivity is the empirical finding that independently trained solutions can often be connected by a low-loss path in parameter space. This suggests that many minima in deep learning are not isolated basins but parts of wider connected regions.
Gradient Noise ScaleA scalar diagnostic that estimates the largest useful batch size by comparing the variance of per-example gradients to the squared norm of the mean gradient: \( B_{\text{noise}} = \operatorname{tr}(\Sigma)/\|g\|^2 \). McCandlish et al. (2018) argue that returns to scaling batch size diminish sharply once \( B \gg B_{\text{noise}} \), giving a principled way to choose batch size during large-scale training.
Adaptive Gradient Clipping (AGC)A per-parameter clipping rule introduced by Brock et al. (2021) that bounds each weight's update by a fraction of the weight's own norm: clip \( g \) so \( \|g\|/\|w\| \le \lambda \). Unlike global-norm clipping, AGC scales naturally with parameter magnitude and made it possible to train Normalizer-Free Networks (NFNets) without batch normalisation while matching its training stability.
Self-Paced LearningA curriculum-learning variant where the model itself decides which examples are 'easy enough' to train on at the current step, by minimising a joint objective \( \sum_i v_i \ell_i - \lambda \sum_i v_i \) over both parameters \( \theta \) and per-example weights \( v_i \in \{0,1\} \) (or \( [0,1] \)). Kumar, Packer & Koller (2010) introduced it as a non-convex EM-style alternative to handcrafted curricula; \( \lambda \) is annealed from low (only easy examples) to high (all examples).
Loss Landscape VisualizationMethods for visualising high-dimensional loss surfaces by projecting parameters onto 1-D or 2-D subspaces. Goodfellow's linear interpolation (2014) plots loss along the line between two solutions; Li et al.'s filter normalisation (2018) plots loss in a 2-D plane spanned by random Gaussian directions normalised per-filter. The latter reveals that residual connections smooth the landscape and that flat minima correspond to wide bowls in the visualisation.
Gradient Surgery (PCGrad) for Multi-Task LearningWhen two task gradients in multi-task learning point in conflicting directions (negative cosine), they partially cancel each other and slow learning. Yu et al.'s PCGrad (2020) projects each task gradient onto the normal plane of any conflicting task's gradient before summing, removing the destructive component. This 'gradient surgery' restores monotone progress on both tasks at modest cost.
Contrastive Learning TheoryThe theoretical account of why contrastive self-supervised objectives like InfoNCE produce useful representations. Wang & Isola (2020) show the InfoNCE loss decomposes into two asymptotic terms — \( \mathcal{L}_{\text{align}} \), pulling positive pairs together, and \( \mathcal{L}_{\text{unif}} \), spreading the marginal feature distribution uniformly on the hypersphere. The downstream linear-probe accuracy correlates almost perfectly with this alignment-uniformity trade-off, giving a geometric explanation for why contrastive learning works at all.
Non-Contrastive SSL (BYOL, SimSiam, Barlow Twins, VICReg)Non-contrastive self-supervised learning learns representations from multiple views of the same example without explicit negative pairs. Methods such as BYOL, SimSiam, Barlow Twins, and VICReg avoid collapse using asymmetry, stop-gradient, redundancy reduction, or variance-preserving terms instead of contrastive negatives.
Disentangled Representation LearningDisentangled representation learning seeks latent coordinates that each correspond to separate underlying factors of variation in the data. It is attractive for control and interpretability, but in the unsupervised setting true disentanglement is usually not identifiable without extra inductive bias or supervision.
Sparse Representations in Deep NetsA sparse representation is one where only a small fraction of units are active for any given input. Deep nets often develop sparsity through ReLU-like nonlinearities or explicit penalties, which can improve efficiency, feature selectivity, and sometimes interpretability.
Representation CollapseRepresentation collapse is the failure mode where many inputs map to nearly the same embedding or hidden state, destroying useful information. It appears in several forms — constant-vector collapse in self-supervision, dimensional collapse where only a few directions survive, and cluster collapse in discrete latents — and each requires a different fix.
Invariance vs EquivarianceA representation is invariant to a transformation if the output does not change when the input is transformed, and equivariant if the output changes in a predictable transformed way. CNN translation equivariance and classifier translation invariance are the canonical example pair.
Embedding GeometryEmbedding geometry studies what information is encoded in the distances, angles, and directions of an embedding space. Properties like similarity, analogy structure, anisotropy, and clustering determine how useful an embedding is for retrieval and downstream tasks.
Metric Learning at ScaleMetric learning at scale trains embeddings so similar items are close and dissimilar items are far apart even when the dataset is too large for naive pairwise or triplet mining. The main challenge is finding informative negatives and keeping computation manageable as the corpus grows.
Representation Alignment Across ModalitiesRepresentation alignment across modalities trains different encoders so paired inputs, such as an image and its caption, land near each other in a shared embedding space. This makes cross-modal retrieval and transfer possible by giving different modalities a common geometry.
Tokenization as RepresentationTokenization is not just preprocessing: it decides which units the model can represent directly and therefore shapes the statistics the model learns. The choice of characters, subwords, bytes, or domain-specific tokens changes sequence length, vocabulary size, inductive bias, and how cleanly concepts map into embeddings.
Autoregressive vs Diffusion TradeoffsAutoregressive models factorise \( p(x) = \prod_t p(x_t \mid x_{<t}) \) and dominate text generation; diffusion models learn a denoising process and dominate continuous-modality generation. The two paradigms differ in likelihood tractability, sampling cost, controllability, and compositionality — and the right choice depends on whether tokens are discrete, parallel decoding is required, and whether log-likelihood or perceptual quality is the figure of merit.
Text-to-Image AlignmentText-to-image (T2I) alignment is the task of making generated images faithfully follow textual prompts — covering spatial layout, attribute binding, count, and style. Modern alignment relies on contrastive image–text encoders (CLIP, SigLIP, T5) injected via cross-attention into a diffusion or flow backbone, plus classifier-free guidance, RLHF-style preference fine-tuning, and reward models that grade prompt adherence.
Generative Model Evaluation (FID, IS, and their limits)Fréchet Inception Distance (FID) and Inception Score (IS) are the standard automated metrics for image generative models; both rely on Inception-v3 features and have well-known biases. Modern T2I evaluation supplements them with CLIPScore, prompt-adherence benchmarks (T2I-CompBench, GenEval), human-preference Elo (ImageReward, HPS), and likelihood / NLL where applicable.
In-Context Learning MechanismsIn-context learning (ICL) is the empirical phenomenon that a frozen LLM solves new tasks from few-shot examples in the prompt. Mechanistic studies show ICL is implemented by a small set of attention circuits — induction heads, function vectors, and implicit gradient-descent-like updates — that emerge during pretraining once the data and depth budget cross a threshold.
Memory in LLMsAn LLM's working memory is the KV-cache for the current context window; longer-term memory is implemented externally by retrieval over vector / hybrid stores, by writing to scratchpads or tool state, or by parameter updates (continual fine-tuning). Each option has a different latency, capacity, and forgetting profile; production systems combine all three.
Alignment Techniques (RLHF, DPO, RLAIF, comparison)Modern LLM alignment uses preference data to adjust a pretrained model so it follows instructions, refuses unsafe content, and ranks desired behaviours above undesired ones. The dominant recipes — RLHF with PPO, DPO and its variants, and RLAIF with AI-generated preferences — share the same Bradley–Terry preference model but differ in optimiser, reward-model dependence, and stability.
Neural Fields / Implicit Neural RepresentationsA neural field represents a continuous signal with a neural network that maps coordinates to values such as color, density, or signed distance. This makes the model itself a compact continuous representation of an image, shape, or scene, with NeRF as the best-known example.
Vision Transformer (ViT) VariantsSince the original ViT, a wide family of variants has emerged that improve data efficiency, locality, hierarchy, and pretraining objective. The most influential are DeiT (training recipe), Swin (windowed hierarchical attention), MAE (masked-image pretraining), DINOv2 (self-distilled features), and SigLIP (sigmoid contrastive pretraining). Each addresses a specific weakness of the vanilla ViT.
Graph TransformersGraph Transformers apply self-attention over graph-structured data, with positional encodings (Laplacian eigenvectors, random walks, shortest-path distances) injecting graph topology that vanilla attention lacks. They generalise message-passing GNNs and have become the leading architecture for molecular property prediction, code understanding, and combinatorial optimisation.
Perceiver ArchitectureThe Perceiver uses cross-attention from a small latent array to a potentially very large input, then performs most computation in latent space. This decouples cost from input length and makes one architecture usable across images, audio, video, and other modalities.
Neural Architecture Search (modern approaches)NAS automates the design of network architectures by searching a parameterised space against a validation objective. Modern NAS abandons the slow RL-controller approach (NASNet) in favour of weight-sharing one-shot supernets (DARTS, ProxylessNAS), zero-cost proxies, and architecture-aware scaling laws (EfficientNet, NFNet). NAS has produced strong vision backbones but plays a smaller role in the LLM era.
Modular Neural NetworksA modular network composes specialised sub-networks (modules) under a routing or composition rule that decides which modules process each input. Mixture-of-Experts is the most successful instance, but the family includes routed adapter networks, modular meta-learners, and compositional architectures designed for systematic generalisation. The motivation is parameter efficiency and reusable skills.
Neural Program InterpretersA neural program interpreter (NPI) is a network that executes program-like computations: looking up arguments, calling sub-routines, manipulating an external memory or stack, and conditioning on intermediate state. Early work (NPI, NTM, DNC, NeuralGPU) targeted symbolic algorithms; modern descendants are tool-using LLMs and chain-of-thought executors that lean on external interpreters and structured memory.
Compressed Sparse Attention (CSA)Compressed Sparse Attention (CSA) is a long-context attention scheme that first compresses the KV cache into block summaries and then performs sparse attention only over the top-k relevant compressed blocks. An added sliding-window branch preserves exact local dependencies, so CSA cuts both KV memory and long-context attention compute without collapsing into a purely local window.
Offline Reinforcement LearningOffline reinforcement learning learns a policy from a fixed logged dataset without further interaction with the environment. Its central difficulty is distribution shift: Bellman backups evaluate actions that are poorly supported by the data, so modern methods either constrain the learned policy to stay near the behavior data or pessimistically down-value unsupported actions.
Model-Based Reinforcement LearningModel-based reinforcement learning learns an explicit or latent model of environment dynamics and uses that model for planning, imagination rollouts, or policy optimization. Its main advantage is sample efficiency, while its main failure mode is model bias: the policy can exploit errors in the learned simulator unless planning and training control compounding prediction error.
Decision TransformersDecision Transformers cast offline reinforcement learning as conditional sequence modeling: a Transformer predicts the next action from past returns-to-go, states, and actions. This avoids explicit Bellman backups and instead treats policy learning like autoregressive imitation conditioned on the desired future return.
Multi-Agent Reinforcement LearningMulti-agent reinforcement learning studies environments where several learning agents interact simultaneously, making each agent's dynamics depend on the evolving policies of the others. The main challenges are non-stationarity, coordination, and credit assignment, which is why centralized training with decentralized execution is a common modern design.
Exploration vs Exploitation (Deep RL View)Exploration versus exploitation is the trade-off between taking actions that seem best under current knowledge and taking actions that improve knowledge of the environment. In deep RL the problem is harder than in bandits because rewards can be sparse, state spaces are large, and short-term randomness is often not enough to discover long-horizon strategies.
Reward HackingReward hacking occurs when an agent maximizes the formal reward signal while failing at the designer's intended objective. It is a general Goodhart-style failure mode in reinforcement learning: stronger optimization pressure often finds loopholes in the proxy reward faster than humans can patch them.
Safe Reinforcement LearningSafe reinforcement learning studies how to optimize long-term return while satisfying safety constraints during training and deployment. The standard formalism is a constrained Markov decision process, where the policy must maximize reward subject to a bound on expected cost, risk, or unsafe-state visitation.
Hierarchical Reinforcement LearningHierarchical reinforcement learning decomposes control across time scales, usually by letting a high-level policy choose skills, options, or subgoals and a low-level policy execute them. This can make sparse-reward and long-horizon problems easier, but only if the learned hierarchy discovers reusable abstractions rather than collapsing back to flat control.
Distributed Training (Data & Model Parallelism)Distributed training scales learning across many devices by splitting either the data, the model, or both. Data parallelism is simplest when the model fits on each device, while model parallel approaches such as tensor and pipeline parallelism are needed when parameters or activations are too large for one accelerator.
Efficient Inference (Distillation, Pruning)Efficient inference reduces latency, memory, or serving cost without retraining a model from scratch, often by distilling a smaller student or pruning unimportant weights and structures. Distillation transfers behavior; pruning removes computation, and the two are often combined with quantization for deployment-grade compression.
Serving LLMs at ScaleServing LLMs at scale is a systems problem of jointly optimizing prompt prefill throughput, token-by-token decode latency, KV-cache memory, batching policy, and fleet utilization. Modern serving stacks rely on continuous batching, prefix caching, PagedAttention, speculative decoding, and sometimes prefill/decode disaggregation to keep both tail latency and GPU cost under control.
Data-Centric AIData-centric AI treats data quality, labeling, coverage, and feedback loops as first-class levers of model performance rather than focusing only on architecture or hyperparameters. The core workflow is to diagnose failure slices, improve the dataset that generates those failures, and measure whether the model gets better for the right reasons.
Dataset Versioning & LineageDataset versioning and lineage track exactly which raw data, labels, transformations, and filters produced a training or evaluation set. They matter because reproducibility, compliance, rollback, and debugging all depend on being able to answer "which data built this model?" with more precision than a folder name or timestamp.
Feature StoresFeature stores are systems for defining, computing, and serving reusable machine-learning features consistently across training and production. Their core promise is point-in-time correctness and train/serve consistency: the feature a model saw offline should match the feature served online for the same entity and timestamp.
Adversarial Robustness (Modern Attacks)Adversarial robustness studies how learned models can be forced into wrong predictions by carefully chosen perturbations that are small, structured, or semantically deceptive. Modern attack families include gradient-based \(\ell_p\) attacks, universal perturbations, adversarial patches, and transfer attacks; defenses must avoid merely hiding gradients while leaving the model fragile.
Distribution Shift & Dataset ShiftDistribution shift occurs when the joint distribution seen at deployment differs from the one used for training or validation. The main cases are covariate shift, label shift, and concept shift; each breaks generalization in a different way and therefore requires different detection and mitigation strategies.
Uncertainty Estimation in Deep LearningUncertainty estimation in deep learning tries to quantify when a model should be unsure, not just what label it predicts. The key distinction is between aleatoric uncertainty from irreducible noise in the data and epistemic uncertainty from limited knowledge of the model, and modern methods such as deep ensembles, MC dropout, and conformal prediction target those uncertainties differently.
ML System Monitoring & Drift DetectionML system monitoring tracks whether a deployed model is still receiving the kind of data it was built for and whether its business and technical behavior remain acceptable. Drift detection is one part of that job: teams also monitor latency, calibration, feature freshness, label delay, feedback loops, and downstream outcomes, because data drift alone does not tell the whole production story.
RLHF as KL-Regularized Policy OptimizationA deeper theoretical view of RLHF treats post-training as optimizing a policy against a learned reward while regularizing toward a reference model with a KL penalty. This viewpoint explains why PPO-RLHF, reward-model training, and even DPO-style objectives are closely related: they are different ways of solving or approximating the same regularized preference-optimization problem.
Law of Total ProbabilityThe law of total probability computes an event probability by summing over mutually exclusive, exhaustive cases. In machine learning it is the basic marginalization identity behind latent-variable models, mixture models, and many Bayesian calculations.
Conditional IndependenceConditional independence means two variables become unrelated once a third variable is known. It is the simplifying assumption that makes graphical models tractable and explains why conditioning can either remove dependence or, in collider structures, create it.
Sufficient StatisticsA sufficient statistic is a summary of the sample that retains all information about a parameter relevant for inference. This is why many classical models can replace an entire dataset with counts, sums, or means without changing the likelihood-based conclusions about the parameter.
Bayes Risk and the Bayes Optimal ClassifierBayes risk is the minimum achievable expected loss under the true data distribution, and the Bayes-optimal classifier attains it by minimizing posterior expected loss for each input. Under ordinary 0–1 loss, that rule becomes “predict the class with highest posterior probability.”
Proper Scoring RulesA scoring rule is proper if a forecaster minimizes expected score by reporting their true predictive distribution. Proper scoring rules matter because they reward honest, calibrated probabilities rather than merely getting the top-ranked class right.
Surrogate Losses and Classification CalibrationSurrogate losses replace hard-to-optimize 0–1 classification loss with tractable objectives such as logistic or hinge loss. A surrogate is classification-calibrated if optimizing it still drives the classifier toward the Bayes-optimal decision rule.
Empirical Risk Minimization (ERM)Empirical risk minimization chooses the model with the smallest average training loss. It is the default principle behind most supervised learning, but it must be paired with capacity control or held-out evaluation because low training loss alone does not guarantee generalization.
Structural Risk Minimization (SRM)Structural risk minimization extends empirical risk minimization by balancing training fit against model complexity. It is the learning-theoretic principle behind regularization, margin control, and choosing among hypothesis classes of different capacity.
Train/Validation/Test SplitA train/validation/test split separates fitting, model selection, and final evaluation into different datasets. The test set is kept untouched until the end so it remains a credible estimate of out-of-sample performance.
k-Fold Cross-Validationk-fold cross-validation rotates a held-out fold through the dataset so every example is used for validation once and training the other times. It uses limited data efficiently for model selection, but it costs multiple training runs and must keep preprocessing inside each fold.
Confusion MatrixA confusion matrix counts predicted labels against true labels. In binary classification it yields the four basic counts—true positives, false positives, true negatives, and false negatives—from which most common thresholded metrics are derived.
Precision-Recall Curve and Average PrecisionA precision-recall curve shows how precision and recall trade off as the decision threshold moves through a ranked list of predictions. Average precision summarizes that curve and is especially informative when the positive class is rare.
Feature Scaling and StandardizationFeature scaling rescales input dimensions to comparable magnitudes, while standardization specifically subtracts the training mean and divides by the training standard deviation. It matters because optimization, distances, and margins can otherwise be dominated by whichever feature uses the largest units.
Hinge LossHinge loss penalizes examples that are misclassified or that lie too close to the decision boundary. It is the convex margin-based loss underlying soft-margin SVMs and emphasizes confident separation rather than calibrated probabilities.
Cost-Sensitive LearningCost-sensitive learning assigns different penalties to different kinds of mistakes instead of treating every error equally. It is the right framework when the real objective is to minimize downstream harm or utility loss rather than raw misclassification rate.
Class Imbalance and ReweightingClass imbalance means some labels are much rarer than others, so an unweighted objective can be dominated by the majority class. Reweighting changes the loss or sampling scheme so rare classes exert more influence during training.
Markov Decision Process (MDP)A Markov decision process formalizes sequential decision-making with states, actions, transitions, rewards, and a discount factor. Its key assumption is that the next-state and reward distribution depends only on the current state and action, which makes Bellman-style planning possible.
Dynamic Programming for RLDynamic programming solves an MDP with a known model by repeatedly applying Bellman updates until values or policies become self-consistent. Policy evaluation, policy improvement, policy iteration, and value iteration are the core algorithms in that family.
Potential Outcomes FrameworkThe potential outcomes framework defines causal effects by comparing the outcomes a unit would have under different treatments. Because only one of those potential outcomes is observed for any given unit, causal inference is fundamentally about identifying missing counterfactuals under defensible assumptions.
Confounding, Colliders, and Simpson’s ParadoxConfounders create misleading associations because they affect both treatment and outcome, while colliders create bias when you condition on them. Simpson’s paradox is the visible symptom that aggregate and stratified associations can reverse direction when the underlying causal structure is ignored.
Attention Is All You Need“Attention Is All You Need” introduced the Transformer: a sequence model built around self-attention instead of recurrence or convolution. The paper mattered because it showed that attention-based, highly parallel sequence modeling could outperform recurrent seq2seq systems and set the template for modern LLMs.
AlexNetAlexNet was the deep convolutional network that won ILSVRC 2012 by a huge margin and triggered the modern deep-learning wave in vision. Its impact came from the full recipe—ImageNet-scale data, GPU training, ReLU, dropout, and augmentation—not from a single isolated trick.
Multiple Hypothesis Testing and False Discovery RateMultiple hypothesis testing asks how to control false positives when many tests are run at once. False discovery rate control, especially the Benjamini–Hochberg procedure, limits the expected fraction of rejected hypotheses that are actually null and is usually less conservative than family-wise error control.
Likelihood Ratio TestsA likelihood ratio test compares how well two nested statistical models explain the same data by taking the ratio of their maximized likelihoods. Large likelihood-ratio statistics indicate that the larger model fits substantially better than the restricted one, and under regularity conditions the test statistic is asymptotically chi-squared.
Importance SamplingImportance sampling estimates an expectation under a target distribution by drawing samples from a different proposal distribution and reweighting them. It is powerful when the proposal places more mass in the important regions of the integrand, but unstable weights can make the variance explode.
Bootstrap Confidence IntervalsBootstrap confidence intervals estimate uncertainty by resampling the observed dataset with replacement and recomputing the statistic many times. They are useful when analytic standard errors are awkward, but they inherit the sample's biases and can fail when the original sample is too small or unrepresentative.
Brier ScoreThe Brier score measures the mean squared error of probabilistic predictions, so it rewards both correctness and calibration. Lower is better, and unlike accuracy it penalizes a confidently wrong 0.99 prediction much more than a cautious 0.6 prediction.
One-Class SVMA one-class SVM learns a boundary around mostly normal data by separating mapped training points from the origin with maximum margin in feature space. It is a classic novelty-detection method because it needs examples of the inlier class but not labeled anomalies.
Anomaly DetectionAnomaly detection identifies observations that look unlikely under the pattern of normal data. The main families are density-based methods, reconstruction-based methods, and one-class classification methods, and the right choice depends on whether you have labels, strong feature engineering, or only normal examples.
Latent Dirichlet Allocation (LDA Topic Models)Latent Dirichlet Allocation models each document as a mixture of latent topics and each topic as a distribution over words. It is a generative model for uncovering coarse semantic structure in bag-of-words corpora, not a modern contextual language model.
Factor AnalysisFactor analysis models observed variables as linear combinations of a small number of latent factors plus variable-specific noise. It is useful when the goal is to explain covariance structure rather than merely reduce dimension, which is the key difference from PCA.
Kalman FilterThe Kalman filter recursively estimates the hidden state of a linear Gaussian dynamical system by alternating a prediction step with a measurement update. It is optimal for that model class because the posterior remains Gaussian and is fully described by a mean and covariance.
Particle FilterA particle filter approximates the posterior over a hidden state with weighted samples, or particles, instead of a single Gaussian. It is useful for nonlinear or non-Gaussian state-space models, but resampling and weight degeneracy are central practical issues.
Canonical Correlation Analysis (CCA)Canonical correlation analysis finds linear combinations of two random vectors that are maximally correlated with each other. It is the right tool when the question is about shared structure between two views of the same examples rather than variance within a single view.
Missing Data and ImputationMissing-data methods try to preserve inference when some values are unobserved by modeling why data are missing and how to fill or integrate over the missing entries. The key distinction is between MCAR, MAR, and MNAR, because imputation is far safer when missingness can be treated as conditionally ignorable.
Target Leakage vs. Data LeakageData leakage is any contamination that lets training or validation use information that would not be available at prediction time. Target leakage is the specific case where features encode the label or a post-outcome proxy for it, so every target leakage problem is data leakage, but not every data leakage problem is target leakage.
Ablation Studies and Experimental ControlAn ablation study removes or alters one component of a system to measure how much that component actually contributes. Experimental control matters because an ablation is only informative when the comparison keeps everything else fixed, including data, tuning budget, and evaluation protocol.
Value IterationValue iteration solves a known Markov decision process by repeatedly applying the Bellman optimality backup until the value function converges. Once the optimal value is approximated, a greedy policy with respect to that value is optimal or near-optimal.
Policy IterationPolicy iteration alternates between evaluating the current policy and improving it by acting greedily with respect to that value function. It often converges in fewer outer loops than value iteration because each improvement step uses a more fully solved subproblem.
Monte Carlo Reinforcement LearningMonte Carlo reinforcement learning estimates values from complete sampled returns rather than from one-step bootstrapped targets. That makes the targets unbiased with respect to the episode return, but usually higher variance than temporal-difference methods.
Credit Assignment ProblemThe credit assignment problem is the problem of determining which earlier actions, states, or internal computations deserve blame or credit for a later outcome. It is hard because rewards and losses are often delayed, sparse, or distributed across many interacting decisions.
Instrumental VariablesInstrumental variables identify causal effects when treatment is confounded, provided an instrument affects treatment, is as-good-as random with respect to unobserved confounders, and influences the outcome only through treatment. In simple linear settings, the IV estimand is the ratio of the instrument-outcome covariance to the instrument-treatment covariance.
CounterfactualsA counterfactual asks what would have happened under a different action or treatment than the one that actually occurred. The central difficulty is that for any individual unit, only one potential outcome is observed, so causal inference always requires assumptions to recover the missing alternative.
Do-CalculusDo-calculus is Pearl's set of graphical transformation rules for turning interventional quantities into observational quantities when the causal graph permits it. It matters because it separates what can be identified from data plus structure from what remains fundamentally unidentifiable.
Bahdanau AttentionBahdanau attention is the original additive attention mechanism for sequence-to-sequence models, where the decoder scores each encoder state before producing the next token. It solved the fixed-context bottleneck of early seq2seq RNNs by letting the decoder look back over the whole source sequence at every step.
Seq2Seq with AttentionSeq2seq with attention augments the encoder-decoder architecture so the decoder conditions on a context vector built from all encoder states at each output step. That change made neural machine translation far more effective than fixed-context seq2seq and directly paved the way to modern cross-attention and Transformer models.
Backpropagation — History (Werbos → Rumelhart/Hinton/Williams)The history of backpropagation is the story of an idea known in pieces before it became a practical neural-network training method. Werbos articulated reverse-mode differentiation for network training in the 1970s, and Rumelhart, Hinton, and Williams turned it into the landmark 1986 demonstration that made multilayer neural networks trainable in practice.
Equivariant Neural NetworksEquivariant neural networks bake a symmetry group action into the architecture so that \( f(g \cdot x) = \rho(g) \cdot f(x) \) holds exactly by construction, achieved by constraining each layer's weights to the operators that commute with the group action. The dominant examples are CNNs (translation), Deep Sets and GNNs (permutation), G-CNNs (discrete rotations), and SE(3)-equivariant networks like EGNN, NequIP, and AlphaFold's IPA (3D rigid motions for molecules and proteins).